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Gravitational Duals from Equations of State II: Large Hierarchies and False Vacua
Raul Jimenez, David Mateos, Pavlos Protopapas, Pau Solé-Vilaró, Pedro Tarancón-Álvarez, Pablo Tejerina-Pérez
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Summary
This paper investigates the reconstruction of holographic duals for strongly coupled quantum field theories (QFTs) in regimes involving large hierarchies and false vacua. Using the gauge/gravity duality, the authors address the 'holographic inverse problem'—reconstructing the bulk scalar potential $V(\phi)$ from boundary thermodynamic data (the equation of state $S(T)$). The study specifically focuses on the Einstein-Klein-Gordon (EKG) model and the challenges posed by the false vacuum regime, such as near-degenerate states and numerical stiffness. The authors improve upon previous Physics-Informed Neural Networks (PINNs) methodologies by developing advanced techniques to accurately reconstruct scalar potentials even when parts of the potential are not directly probed by the input data, successfully bridging the gap between holography and machine learning.
Entities (7)
Relation Signals (4)
Einstein-Klein-Gordon (EKG) model → contains → Scalar Potential
confidence 100% · the gravitational theory is an Einstein-Klein-Gordon (EKG) model in five dimensions... the scalar potential V(ϕ) of the EKG model.
Gauge/Gravity Duality → relates → Quantum Field Theories
confidence 100% · Holography maps the quantum properties of strongly coupled gauge theories... to the classical properties of gravitational theories.
Physics-Informed Neural Networks (PINNs) → usedtosolve → holographic inverse problem
confidence 100% · Building on previous work based on Physics-Informed Neural Networks (PINNs), we extend the holographic inverse problem...
False Vacuum → characterizedby → Large Hierarchies
confidence 90% · regimes characterized by large hierarchies and the presence of false vacua.
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Abstract
Abstract:We investigate the reconstruction of holographic duals for strongly coupled quantum field theories in regimes characterized by large hierarchies and the presence of false vacua. Within the gauge/gravity duality, these features translate into non-trivial thermodynamic behaviour and exotic renormalization group flows, including skipping flows between non-adjacent fixed points. Building on previous work based on Physics-Informed Neural Networks (PINNs), we extend the holographic inverse problem of reconstructing the bulk scalar potential from boundary thermodynamic data into this new regime. This setting presents a variety of conceptual and numerical challenges, such as near-degenerate states, large hierarchies of energy scales, and regions of the potential that are not directly probed by the input data. We develop a set of methodological advances that overcome these obstacles, thereby improving the established PINNs-based methodology and extending it to new physical regimes of interest that were previously out of reach. Applying the developed framework, we demonstrate accurate reconstruction of scalar potentials deep into the false vacuum regime, achieving robust agreement with the physical features of the underlying thermodynamics despite significant numerical stiffness. Our results extend the bridge between holography and machine learning, and suggest that data-driven approaches can provide new insights into the structure of strongly coupled systems.
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- Source: https://arxiv.org/abs/2606.30117v1
- Canonical: https://arxiv.org/abs/2606.30117v1
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aainstitutetext: Departament de Física Quántica i Astrofísica, Universitat de Barcelona, Martí i Franquès 1, ES-08028, Barcelona, Spain.bbinstitutetext: Institut de Ciències del Cosmos (ICC), Universitat de Barcelona, Martí i Franquès 1, ES-08028, Barcelona, Spain.ccinstitutetext: Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig Lluís Companys 23, ES-08010, Barcelona, Spain.ddinstitutetext: Institute for Applied Computational Science, Harvard University, Cambridge, MA. Gravitational Duals from Equations of State I: Large Hierarchies and False Vacua Raul Jimenez a,b,c David Mateos d Pavlos Protopapas a,b Pau Solé-Vilaró a,b Pedro Tarancón-Álvarez a,b Pablo Tejerina-Pérez raul.jimenez@icc.ub.edu; dmateos@fqa.ub.edu; pprotopapas@g.harvard.edu; pau.sole@fqa.ub.edu; pedro.tarancon@fqa.ub.edu; pablo.tejerina@icc.ub.edu Abstract We investigate the reconstruction of holographic duals for strongly coupled quantum field theories in regimes characterized by large hierarchies and the presence of false vacua. Within the gauge/gravity duality, these features translate into non-trivial thermodynamic behaviour and exotic renormalization group flows, including skipping flows between non-adjacent fixed points. Building on previous work based on Physics-Informed Neural Networks (PINNs), we extend the holographic inverse problem of reconstructing the bulk scalar potential from boundary thermodynamic data into this new regime. This setting presents a variety of conceptual and numerical challenges, such as near-degenerate states, large hierarchies of energy scales, and regions of the potential that are not directly probed by the input data. We develop a set of methodological advances that overcome these obstacles, thereby improving the established PINNs-based methodology and extending it to new physical regimes of interest that were previously out of reach. Applying the developed framework, we demonstrate accurate reconstruction of scalar potentials deep into the false vacuum regime, achieving robust agreement with the physical features of the underlying thermodynamics despite significant numerical stiffness. Our results extend the bridge between holography and machine learning, and suggest that data-driven approaches can provide new insights into the structure of strongly coupled systems. 1 Introduction Holography Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj ; aharony2000large maps the quantum properties of strongly coupled gauge theories with a large number of colours to the classical properties of gravitational theories in one higher dimension. Under this map, the thermodynamic properties of the gauge theory are encoded in properties of black hole horizons in the dual gravitational solutions. In particular, the equation of state of the field theory, encoded in the thermodynamic relation between entropy density and temperature, S(T)S(T), is determined by the set of all static black hole solutions. Nevertheless, the most far-reaching consequence of the duality is arguably that even arbitrarily-far-from-equilibrium dynamics of the gauge theory can be determined by evolving the dual Einstein’s equations in time. In such dynamical regimes, holography often provides the only available framework for controlled first-principle calculations, in contrast to equilibrium settings where alternative approaches, such as lattice gauge theory, can sometimes be employed. Hence, an appealing strategy is to construct a holographic model that reproduces the equilibrium properties of a gauge theory of interest, and then use the gravitational dual to access the out-of-equilibrium dynamics. However, implementing this strategy requires solving a challenging inverse problem: given the equilibrium equation of state of the gauge theory one must determine the dual gravitational theory. In a previous paper Bea:2024xgv , we presented a novel approach to solve this inverse problem based on Physics-Informed Neural Networks (PINNs), focusing on the case where the gravitational theory is an Einstein-Klein-Gordon (EKG) model in five dimensions. Our algorithm reconstructed not just a specific black hole solution but the whole gravitational theory itself, encoded in the scalar potential V(ϕ)V(φ) of the EKG model. This was successfully demonstrated for a family of four-dimensional Conformal Field Theories (CFTs) deformed by a relevant operator of conformal dimension Δ=3 =3, whose dual potential was parametrised by a single parameter ϕM _M that hence controlled the nature of the CFT thermal phase transition. The method was applied to theories exhibiting crossovers (ϕM=5 _M=5), second-order (ϕM=1.08 _M=1.08) and first-order (ϕM=1 _M=1) phase transitions, achieving sub-percent relative errors in the recovered potential. In tarancon2025efficient , the framework was further developed through the introduction of novel features in the neural network (N) pipeline, such as multi-head training and unimodular regularisation techniques, that improved the efficiency of the transfer learning process across different parameter regimes. However, the approaches used in both aforementioned works proved insufficient to accurately recover potentials with values of ϕM≲0.8 _M 0.8. As was shown in bea2018heating , for values of ϕM _M below a critical threshold ϕM≃0.5808 _M 0.5808, due to its functional form the scalar potential develops additional features in the form of two new local extrema between its original global maximum and minimum. This has important implications across various aspects in the problem at hand. On the physics side, potentials below this critical threshold are dual to boundary theories that posses a false vacuum, namely a zero-temperature, Lorentz-invariant state with energy density higher than that of the true vacuum. A comparison between such a boundary theory and another one without a false vacuum can be seen in Fig. 1. Without loss of generality, we will assign zero energy to the true vacuum, so the energy of the false vacuum will be positive. In the holographic context this can be achieved by an appropriate choice of counterterms. The gravitational theories dual to these QFTs can give rise to so-called exotic renormalization group (RG) flows when looking at the behaviour of the running coupling at the boundary bea2018heating ; kiritsis2017exotic . For instance, we can find “skipping flows” that connect non-adjacent fixed points while bypassing intermediate CFTs, which is unusual in field theory, where flows stop at the first infrared (IR) fixed point available. Finally, the thermodynamics at the boundary become remarkably non-trivial, with the equation of state developing a complex multi-valued structure that encodes information about all the intermediate CFTs in the potential landscape. (a) (b) (c) (d) Figure 1: Entropy density (a), energy density (b), and entropy-to-temperature cubed ratio (c) as a function of temperature for two different values of the parameter ϕM _M. The value ϕM=0.8 _M=0.8 corresponds to a first-order phase transition. As we decrease the value of this parameter, the transition becomes more pronounced and develops a false vacuum state, as seen for the case of ϕM=0.55 _M=0.55. At T/Λ=0T/ =0 there are two available states: the false vacuum and the true vacuum states, as seen in the dashed blue lines in (b). In (d) we plot the value of the Ricci scalar in the bulk, evaluated at the horizon and normalized by its asymptotic AdS value, as a function of the scalar field at the horizon. The red and blue dots represent the true and false vacua, respectively. The gap in the blue dashed curve reflects the absence of solutions in that region. In both theories, a large separation of scales between their respective vacua can also be observed. On the computational side, the false vacuum regime presents qualitatively new challenges that go beyond mere numerical stiffness. As will be explained in further detail below, the scalar potentials in this regime will now contain a region whose dual thermal states in fact belong to a different boundary theory and are therefore absent from the equation of state given as an input to the PINNs model, as seen pictorally in Fig. 1(d). Additionally, the false and true vacuum states correspond to the same point (the origin) in the (S,T)(S,T) plane (see Fig. 1(a)) despite corresponding to very different bulk solutions, creating a degeneracy that must be resolved. Finally, there is a large separation of scales between the defining features of the potential; the local extrema are shallow compared to the deep global minimum, and hence represent small-scale features present in the solution (see Figs. 2 and 1(d)). This issue in particular is well-known in the differential equation (DE) mathematical and numerical literature hairer1996solving ; Seiler2025 , and in our context is prevalent even in simpler cases outside of the false vacuum regime. It is presumably one of the main reasons why the PINNs-based strategy used in previous works had so far not yielded satisfactory results when applied to the near-false-vacuum regime of 0.5808≤ϕM≤0.80.5808≤ _M≤ 0.8, where one might hope it could still be applicable. Some of the challenges discussed above stem in part from the fact that the scalar potential used to benchmark our algorithm is derived from a globally well-defined superpotential. Relaxing this assumption could mitigate certain features, such as the pronounced hierarchy between the false and true vacua. However, rather than simplifying the setup, we choose to treat these difficulties as a virtue, using them to probe the limits of the reconstruction framework. Building on the previous successes achieved using PINNs in this holographic inverse problem Bea:2024xgv ; tarancon2025efficient , a natural next step is to extend the methodology to access these more intricate and physically rich regimes. In this work, we address the challenges outlined above through a series of targeted and notable improvements to the PINNs-based algorithm. The rest of the work is organised as follows. In Section 2, we review the holographic model including the functional form of the potential used and the formulation of the direct and inverse problems, with an emphasis on the new theoretical and technical aspects that arise in the false vacuum regime. The methodology used is presented in Section 3, including the architectures and novel techniques employed, as well as the training procedures. In Section 4, we present the results of the potential reconstruction for two cases in the regimes of interest, namely ϕM=0.8 _M=0.8 and ϕM=0.55 _M=0.55. Finally, results and an outlook on future directions are discussed in Section 5. Our full code is publicly available in the following GitHub repository: NNHolo. 2 Holographic model 2.1 The potential As outlined in the previous section, in this work we continue to explore the bottom-up gravity model which had been successfully recovered, within a certain parameter regime, from the thermodynamics of the dual boundary theory in our earlier work Bea:2024xgv . The action of the five-dimensional EKG model takes the form S=2κ52∫x5−g[14R−12(∇ϕ)2−V(ϕ)],S= 2 _5^2 dx^5 -g [ 14R- 12(∇φ)^2-V(φ) ]\,, (1) where κ5 _5 is the gravitational coupling, which hereafter we will set to κ52=2 _5^2=2. V(ϕ)V(φ) is the scalar potential, derived from a so-called superpotential W(ϕ)W(φ) such that V(ϕ)=−43W(ϕ)2+12W′(ϕ)2.V(φ)=- 43W(φ)^2+ 12W (φ)^2. (2) Here, it is important to note that although any extremum of W(ϕ)W(φ) is necessarily an extremum of V(ϕ)V(φ), the reverse does not generally hold. In particular, extrema of both the potential and superpotential will be dual to supersymmetric CFTs, while extrema of the potential but not of the superpotential will be dual to non-supersymmetric CFTs.111Assuming that our bosonic theory is the truncation of a supersymmetric theory. Following bea2018heating , we choose the superpotential W(ϕ)=−32−ϕ22−ϕ44ϕM2+ϕ610,W(φ)=- 32- φ^22- φ^44φ^2_M+ φ^610, (3) where we have set the asymptotic AdS radius to unity. This is a particular choice within the formalism at hand, which has been used in a variety of works to connect Einstein’s equations with the holographic RG de2000holographic ; kiritsis2017exotic ; kiritsis2014holographic . The value of the parameter ϕM _M determines the form of the potential and hence the properties of the dual boundary theory, such as its RG flow or its thermal structure. For ϕM>1.08 _M>1.08, the thermal physics exhibits a crossover between two conformal regimes controlled by an ultraviolet (UV) and an IR fixed point; as we lower the parameter, this becomes a 2nd2^nd-order phase transition at ϕM≃1.08 _M 1.08, and then a 1st1^st-order phase transition for ϕM≲1.08 _M 1.08. For these cases, the potential has the usual form involving one maximum and one minimum (in the region of interest), both of which are also extrema of the superpotential (see Fig. 2(a)), each corresponding to AdS solutions of different radii. The phase transition becomes more pronounced as ϕM _M decreases further, entering the near-false-vacuum regime below ϕM=0.8 _M=0.8. Eventually we reach a point at ϕM≃0.5808 _M 0.5808 where the potential develops a degenerate critical point at ϕ≃0.9015φ 0.9015, at which both the first and second derivatives of V(ϕ)V(φ) vanish. Beyond this value, we enter the false vacuum regime where the potential develops two additional features in the form of a local minimum and maximum between the two original extrema. These new critical points are extrema of the potential but not of the superpotential. The true vacuum of the gauge theory is associated to an RG flow in the bulk that starts at the original maximum and ends at the original minimum. The false vaccum is associated to a flow from the original maximum to the new local minimum. The general form of these two possible morphologies of the potential, with the global and local stationary points, is shown in Fig. 2. (a) (b) Figure 2: The form of the potential V(ϕ;ϕM)V(φ; _M) for different choices of the parameter ϕM _M. On the left, the potential for ϕM=0.8 _M=0.8 corresponds to a QFT at the boundary with a 1st1^st-order phase transition (a). On the right, we enter the false vacuum regime with ϕM=0.55 _M=0.55 as the potential develops an additional local minimum and maximum in the zoomed-in region (b). Notice the large separation of scales between features of the potential in (b), as well as the order of magnitude increase in the range of the potential when compared to (a). The parameter ϕc _c is found numerically and will be introduced in Section 2.2. 2.2 The direct problem In the context of the gauge/gravity duality, the scalar potential V(ϕ)V(φ) in the bulk action (1) encodes all characteristics of the gauge theory, including its thermodynamic behaviour. To extract the thermodynamics from a given potential requires solving the so-called direct problem, which involves finding every static, smooth black hole configuration that satisfies the EKG field equations. To this end, we take the following ansatz for the bulk metric in Eddington-Finkelstein coordinates: ds2=−Adt2+Σ2(dx2+dy2+dz2)−2u2dtdu,ds^2=-Adt^2+ ^2(dx^2+dy^2+dz^2)- 2u^2\,dtdu\,\,, (4) where t,x,y,z\t,x,y,z\ are boundary coordinates and u is the null holographic coordinate such that A, Σ , and the bulk scalar field ϕφ are functions of u. The AdS boundary is located at u=0u=0. Using this ansatz to solve the bulk equations of motion derived from (1), we obtain the following ordinary differential equations (ODEs): Σ′+2uΣ′+23Σϕ′2 + 2u + 23 φ 2 = = 0, 0\,\,, (5a) A′+83u4V(ϕ)+A′(2u+3Σ′Σ) A + 83u^4V(φ)+A ( 2u+ 3 ) = = 0, 0\,\,, (5b) ϕ′−1u4A∂V(ϕ)∂ϕ+ϕ′(2u+A′A+3Σ′Σ) φ - 1u^4A ∂ V(φ)∂φ+φ ( 2u+ A A+3 ) = = 0, 0\,\,, (5c) A′+2AΣ′Σ−2Σ3Σ′(Aϕ′2−2u4V(ϕ)) A +2A - 2 3 (Aφ 2- 2u^4V(φ) ) = = 0, 0\,\,, (5d) where a prime indicates differentiation with respect to u. To solve the equations of motion we must fix the sole input in the equations, namely the potential V(ϕ)V(φ), and impose appropriate boundary conditions (BCs) that ensure that near the boundary at u=0u=0 the geometry is asymptotically AdS. Hence, the near-boundary behaviour of the bulk functions is: A(u) A(u) = = 1u2+⋯, 1u^2+·s\,\,, (6a) Σ(u) (u) = = 1u+⋯, 1u+·s\,\,, (6b) ϕ(u) φ(u) = = Λu+⋯. u+·s\,\,. (6c) We have set the coefficients of the leading terms in A and Σ to unity without loss of generality, and the dots indicate subleading terms in the limit u→0u→ 0. The boundary condition for the scalar field ϕφ imposes the fact that the CFT is deformed by a relevant operator O with dimension Δ=3 =3 and source Λ . The latter is the only intrinsic scale in the resulting QFT, and we will set Λ=1 =1 hereafter. For any value of the scalar field at the horizon, ϕ(u=uH)≡ϕHφ(u=u_H)≡ _H, imposing these asymptotic functional forms leads to a unique pair A1,Σ0\A_1, _0\ of leading-order coefficients related to the surface gravity and the area density of the black brane horizon, from which the temperature T and entropy density S can be readily calculated: T T = = A14π, A_14π\,\,, (7a) S S = = πΣ03. π _0^3\,\,. (7b) Thus, we can construct the thermodynamic curve S(T)S(T) associated to thermal states at the boundary by numerically integrating the EKG equations (5) from the horizon to the boundary for each value of ϕH _H, such that this parameter takes on the role of indexing the set of solutions. This direct problem can be easily carried out once V(ϕ;ϕM)V(φ; _M) has been specified, highlighting the key role of the potential in the setup. It is important to note that although in theories where a phase transition is present there will be multiple states (S,T)(S,T) with the same T, it still remains true that each will be uniquely associated to a value of ϕH _H. However, as we enter the false vacuum regime by lowering the value of the free parameter to ϕM<0.5808 _M<0.5808, we begin to depart from this standard picture. Indeed, these potentials will now contain a range of values of ϕH _H whose solutions do not correspond to thermal states of the original QFT defined by the maximum at ϕ=0φ=0, but to thermal states of the QFT defined by the new local maximum in the potential. The values of ϕH _H that do not map to thermal states of the original QFT range from the new local minimum at ϕ=ϕFVφ= _FV to some critical point ϕ=ϕcφ= _c that lies passed the new maximum, as illustrated in Fig. 2(b). Nevertheless, the form of the potential within this range of values of the scalar field still affects some solutions that do correspond to thermal states of the original QFT. The reason is as follows. From ϕc _c until the global minimum of the potential, the solutions to the direct problem once again correspond to thermal states of the S(T)S(T) curve. To obtain these solutions one must integrate the EKG equations from ϕH>ϕc _H> _c back to ϕ=0φ=0. Since this integration region includes the region (ϕFV,ϕc)( _FV, _c), this part of the S(T)S(T) curve is sensitive to the value of the potential in the range of values of ϕH _H that does not give rise to thermal states of the original QFT. In this theory, solutions with ϕH _H corresponding to either of the two minima or to ϕc _c all correspond to states with zero entropy and temperature. The degenerate nature of these points, along with the portion of the potential that corresponds to thermal state solutions of a different theory, significantly increases the difficulty of solving the direct problem numerically by integrating the equations. 2.3 The inverse problem The inverse problem consists of recovering the bulk theory given only the boundary data; that is, recovering the scalar potential V(ϕ)V(φ) characterising the action (1) that is dual to a known equation of state S(T)S(T). As we have seen, each pair of values (T,S)(T,S) in the thermodynamic curve represents a thermal state making up the boundary data of the QFT, and each state is dual to a planar black hole solution in the bulk with the same entropy density and temperature. Then, as done in the direct problem, a given pair of values for (T,S)(T,S) sets boundary values (at u=uH=1u=u_H=1) for the metric functions A(u)A(u) and Σ(u) (u) in (4). The relation is given in Eqs. (7a, 7b). Formulating the problem in this way, the potential V(ϕ)V(φ) is an unknown, free function of the scalar field ϕφ, which in turn is one of the three functions that appear in the EKG equations (5). The challenge now becomes inverting this free function V(ϕ)V(φ), or in other words, finding V(ϕ)V(φ) such that the ODEs admit solutions for all the BCs set by the different (S,T)(S,T) values. Evidently, this inversion of V(ϕ)V(φ) has to be done in parallel to finding solutions for Σ(u),A(u) (u),A(u) and ϕ(u)φ(u) that fulfill each of the different BCs. As one might expect, there will be as many solutions as BC instances (or equivalently, points in which the S(T)S(T) curve is sampled), but all of them need to share the same free function V(ϕ)V(φ) in the ODEs. This strong requirement, along with the fact that the ODEs at hand are coupled and highly non-linear, are in part what makes the inverse problem so numerically challenging, and what motivated the adoption of a PINNs framework to tackle it in earlier works. Moreover, this impracticability of using traditional numerical methods is especially true for recovering potentials in the false vacuum regime, where part of said potential will not be directly encoded in the initial data; as a result, in those cases any form of iterative numerical approaches that rely on prior solutions, such as certain applications of spectral or relaxation methods, will be effectively rendered out of use despite their general suitability and accuracy in solving other equally difficult numerical problems. In principle, there is no mathematical guarantee that there exists such a function, or that there are no degeneracies related to it. However, one may expect a unique solution to exist for physically-motivated S(T)S(T)’s, and this is plausible mathematically since we wish to constrain one function by specifying another. To simplify the problem, in this paper we will make the mild assumption that S(T)≈T3S(T)≈ T^3 both at high and low temperatures, as corresponds to a QFT with an RG flow between an UV and an IR fixed points. 3 Methodology In this section, we outline the methodology employed to reconstruct the potential and solve the EKG equations using PINNs. We begin by reviewing the relevant background, followed by introducing the specific equations to be solved and describe their formulation within this framework. We then discuss the machine learning (ML) implementation of the problem in the context of PINNs. Finally, we highlight several technical modifications introduced in our approach which depart from the standard PINN setup. 3.1 Background PINNs were first introduced in the pioneering works of Dissanayake and Phan-Thien dissanayake_neural-network-based_1994 and subsequently refined by Lagaris et al. lagaris_artificial_1998 ; lagaris_neural-network_2000 . Since then, PINNs have been established as a versatile framework that can be used to solve partial and ordinary differential equations (PDEs/ODEs). Their efficacy in addressing a broad spectrum of challenging physical problems was rigorously demonstrated by Raissi et al. raissi2019physics , while further methodological and applied advancements have been reported in mattheakis2021hamiltonian ; sirignano2018dgm ; zhu2019physics . Within this paradigm, each unknown function in the governing equations is typically represented by a dedicated neural network, most commonly implemented as a multilayer perceptron (MLP) consisting of stacked, fully-connected layers interleaved with non-linear activation functions. The training procedure consists of minimizing a loss functional constructed from the squared residuals of the differential equations, thereby embedding the governing physical laws directly into the optimisation objective. For alternative strategies used to incorporate physical constraints into ML models, the reader is referred to Choudhary2019 ; Choudhary2020 ; Greydanus2019 . Although PINNs still generally rank below traditional numerical methods when it comes to computational efficiency and accuracy, they present novel benefits. For instance, they offer closed-form approximations to solutions, thereby avoiding the reliance on traditional iterative solvers and significantly reducing computational costs. They are mesh-free, meaning that they enable on-demand evaluation of solutions once training is complete, which is particularly advantageous in the treatment of complex systems. Moreover, their capacity to exploit transfer learning (TL) facilitates the rapid identification of new solutions by leveraging knowledge acquired from related previous tasks, thus improving adaptability and accelerating convergence. An additional strength of PINNs lies in their invertibility, making them well-suited for inverse problems where reconstructing inputs from observed outputs is required. Furthermore, the framework can be extended to include equation parameters as inputs, allowing for a unified treatment of both state variables and system parameters within the same architecture. One crucial element of our setup are the so called “solution bundles”. This technique was first developed in flamant2020solving as an extension of Lagaris’ original method lagaris_neural-network_2000 . This consists of training the neural network (N) on a variety of solutions at once, usually parametrized by different initial/boundary conditions or constants appearing on the equations themselves. As a result, the trained network can be reused more effectively, enabling faster execution of tasks that involve evaluating numerous solutions, including Bayesian parameter inference, uncertainty propagation in dynamical systems, and various classes of inverse problems. 3.2 Einstein-Klein-Gordon equations We will now write the EKG equations (5) in a more suitable way for the PINNs pipeline. First, we will redefine the unknown functions A(u)A(u), Σ(u) (u), ϕ(u)φ(u) such that they are bounded in the entire computational domain u∈[0,1]u∈ [0,1 ]. We can remove the singular behaviour near u=0u=0 seen in expression (6) by introducing the following field redefinitions: Σ~=uΣ,A~=u2A. =u\, \,\,,~~~~ A=u^2A\,. (8) From a computational point of view, it is more efficient for the PINNs setup to deal with first-order ODEs at most. To this end, the system of equations (5) can be rewritten by introducing A~,Σ~ A, and ϕφ as independent variables. Together with the constraint (5d), this results in six functions Σ~(u),A~(u),ϕ(u),νΣ(u),νA(u),νϕ(u)≡ψ→(u)\ (u), A(u),φ(u), _ (u), _A(u), _φ(u)\≡ ψ(u) subject to the following seven coupled first-order ODEs Eα=0,α=1,…,7,E_α=0\,, α=1,…,7\,, (9) where E1 E_1 = = νΣ−Σ~′, _ - \,, (10a) E2 E_2 = = νA−A~′, _A- A \,, (10b) E3 E_3 = = νϕ−ϕ′, _φ-φ \,, (10c) E4 E_4 = = νΣ′+23Σ~νϕ2, _ + 23 \, _φ^2\,, (10d) E5 E_5 = = u2Σ~νA′+83V(ϕ)Σ~+νA(3u2νΣ−5uΣ~)+A~(8Σ~−6uνΣ), u^2\, \, _A + 83\,V(φ)\, + _A\, (3u^2\, _ -5u\, )+ A (8 -6u\, _ )\,, (10e) E6 E_6 = = u2Σ~A~νϕ′max(νϕ,10−3)−Σ~max(νϕ,10−3)dVdϕ+(−3uA~Σ~+u2Σ~νA+3u2νΣA~), u^2\, \, A\, _φ max ( _φ,10^-3 )- max ( _φ,10^-3 )\, dVdφ+ (-3u\, A\, +u^2\, \, _A+3u^2\, _ \, A )\,, (10f) E7 E_7 = = (uνΣ−Σ~)(u2Σ~νA+2u2A~νΣ−4uA~Σ~)−23uΣ~2(u2A~νϕ2−2V(ϕ)). (u\, _ - ) (u^2\, \, _A+2u^2\, A\, _ -4u A )- 23u\, ^2 (u^2 A\, _φ^2-2V(φ) )\,.\,\,\,\,\,\, (10g) Here we can indeed observe the free function V(ϕ)V(φ) along with its first derivative, such that for any given V(ϕ)V(φ) one would obtain different solutions ψ→(u) ψ(u). Note that E6E_6 is the only equation in which there are terms being divided. This was done in order to avoid the approximate trivial solution to which the PINNs setup was observed to converge to during training tests, which was to set νϕ=0 _φ=0. In particular, the model was finding that it was beneficial in terms of reducing the loss function to set this function to zero, even if this solution was wrong, as this then reduced E6E_6 to a much simpler equation to solve. The choice between νϕ _φ and 10−310^-3 ensures that the division does not blow up if the model tries to set a low value for νϕ _φ. The BCs discussed in Sec. 2 can be written as follows A~|u=0 A|_u=0 =1, =1\,\,, (11a) Σ~|u=0 |_u=0 =1, =1\,\,, (11b) ϕ|u=0 φ|_u=0 =0, =0\,\,, (11c) νϕ|u=0 _φ|_u=0 =1, =1\,\,, (11d) A~|u=1 A|_u=1 =0, =0\,\,, (11e) νA|u=1 _A|_u=1 =−4πT, =-4π T\,\,, (11f) Σ~u=1 _u=1 =(S/π)1/3. = (S/π )^1/3\,\,. (11g) These conditions are imposed by a combination of the behaviour the metric functions must follow at the boundary (AdS asymptotics at u=0u=0, see (6)) and at the horizon at u=1u=1, properties of the boundary theory such as its scale and the conformal dimension of its operator deforming the UV fixed point, and the relationship between the metric functions at the horizon and the thermal states at the boundary given in (7). In summary, in the direct problem we choose a value of the scalar field at the horizon ϕH _H, we obtain the numerical solution of the black brane, and we read off the entropy and the temperature. In contrast, in the inverse problem we start from the equation of state S(T)S(T) and use it as a set of BCs to try to recover a potential V(ϕ)V(φ) that fits them all simultaneously. 3.3 PINNs implementation In the following subsections we will present the PINNs-based strategy that was used to improve on the results from our previous work and to probe the false vacuum regime. We will take the same base setup that worked for boundary data corresponding to crossovers and to 2nd2^nd-order and 1st1^st-order phase transitions, and extend it with a number of novel techniques and modifications. These improvements ultimately allow the NNs to overcome the significant difficulties introduced in previous sections associated with inverting potentials in the novel physical regimes we wish to explore. For extensive details on general aspects of the original setup, such as sampling of the points on the S(T)S(T) or the independent variable generator, we refer the reader to Bea:2024xgv . Below, we will simply present the improvements and novel techniques used on top of the aforementioned base setup, highlighting the reasons why they are necessary to probe the false vacuum regime. 3.3.1 Architecture and activations The architecture used for this problem is shown in Fig. 3. The pipeline is comprised of two different NNs. The first one, N-Solver, is composed of 6 different nets, each one outputting one of the 6 bulk metric function solutions to the DEs in (10) as outputs, that is, the functions A~,Σ~,ϕ A, ,φ and their corresponding first-order derivatives νA,νΣ,νϕ _A, _ , _φ. As inputs to these nets, along with the independent variable u and the boundary data (T,S)(T,S), we will also give a novel input parameter Z to the nets that characterizes the position of each particular sampled point along the S(T)S(T) that is used as an input. The numerical computation carried out to obtain this parameter is explained below in subsection 3.3.2. In order to give more freedom to the nets in these stiffer regimes, these will have 5 hidden layers with 128 neurons each. The second network, N-V, will be solving for the scalar potential V(ϕ)V(φ), taking as input the solution ϕ(u)φ(u) coming from the N-Solver nets. Once again, now this net has been given more parameters in the form of 5 hidden layers with 64 neurons each. This is particularly important in the cases attempted in this paper, as the aforementioned large separation of scales between the extrema of the potentials to be recovered (see Fig. 2) requires an enhanced freedom in the nets if one hopes to accurately capture the defining physical features. However, it is equally important to note that over-increasing the number of available parameters in NNs can lead to overfitting, where the model additionally learns noise and outliers in the data such that it performs poorly on unseen data later on. This particular architecture presented here was determined through trial and error within the explored regime of the inverse problem at hand, adapting proven configurations from the inverse problems attempted in our previous study. In summary, we have: [N-Solver]i:[128,128,128,128]i, with i=1,…,6; [N-Solver]_i:[128,128,128,128]_i, with i=1,…,6; N-V:[64,64,64,64,64]. 31.29802ptN-V:[64,64,64,64,64]. As previously done, the activation functions used will be the tanh(x)(x) for N-Solver nets and the SiLu function for the N-V net, where SiLu(x)=x(1+e−x)−1(x)=x(1+e^-x)^-1. Figure 3: Neural Network setup. Novel features include the affine input parameter Z, Gaussian localization in Z in the N-Solver nets and a setup with larger, deeper nets. 3.3.2 Affine parameter A novel key feature of our implementation is the addition of the affine parameter of the boundary S(T)S(T) curve. There are two main advantages that come with introducing this parameter; firstly, it allows us to break the degeneracy between the false and the true vacuum points in the curve and, secondly, it makes it possible to sample the curve uniformly. This in turn allows the PINNs setup to better learn how the obtained solutions change as one moves along the S(T)S(T) input parameters. As was previously explained in Section 2.1, the QFTs whose dual potential we wish to recover are characterized by the presence of a state that has a finite energy density for a vanishing temperature, a so-called false vacuum, as opposed to the true vacuum state of the theory in which the energy density and the temperature vanish simultaneously (see Fig. 1). However, when one translates energy density to entropy density, both the false vacuum and the true vacuum of the theory are mapped to the point S=0S=0 and T=0T=0. Despite the fact that these two points are very close to each other in phase space, the solutions to the ODE system are very different. This makes the feature problematic for the PINNs setup, since the unknown functions are NNs that depend parametrically both on the entropy and the temperature and hence will be unable to break the degeneracy between the false and the true vacuum. By introducing the affine parameter of the S(T)S(T) curve as an additional input to the N model, the false vacuum will posses a smaller value for the affine parameter than the true vacuum, therefore breaking the degeneracy between these two points when feeding them as inputs to the model. Furthermore, as mentioned earlier the second advantage given by the affine parameter is that it allows us to uniformly sample the points along S(T)S(T). Once this parameter is computed, one can choose a set of evenly-spaced points along the affine parameter direction. In this way, we ensure that each region of the curve has the same weight in terms of number of points contained when training the model. The numerical computation of the affine parameter is quite standard. We start with a list of points Ti,Si\T_i,S_i\. In order to compute it, we first need to choose a particular parametrization of the curve T(i),S(i)\T(i),S(i)\. We use the positional index as an auxiliary parameter of the curve. With this, we can interpolate both the temperature and the entropy as a function of this parameter. Once this is done, the affine parameter can be computed as the norm of the tangent vector to the curve Z(s)=∫0s(dTdi)2+(dSdi)2i,Z(s)\,=\, _0^s ( dTdi )^2+ ( dSdi )^2\,di, (12) where the integration constant is chosen such that the high temperature points correspond to small values of the affine parameter. Finally, this parameter is normalised to Z∈[0,1]Z∈[0,1] and is uniformly sampled in this interval. This sampling will generate an evenly distributed sampling of points T(Z),S(Z)\T(Z),S(Z)\ which will then be given to the neural network as inputs, together with the independent variable u and the affine parameter Z. 3.3.3 Gaussian localization A key technique to approach the inverse problem at hand, Gaussian localization (GL), was introduced in our previous work Bea:2024xgv . We will now first review this aspect, providing novel insight on the intuition behind it, and will then detail how it was further implemented on the aforementioned affine parameter, as displayed in Fig. 3. The GL technique gives the N the ability to capture a sense of locality in some input parameter to the net (in our case, the independent variable or the affine parameter). Specifically, it allows for a tuning of the parameters of the N that introduces modifications of the solution only in certain regions around a specific value of the input parameter while leaving solutions outside of this region almost unchanged. The GL technique is implemented as follows. First, one selects the input parameter x that will undergo GL. Then, a gaussian window centered at μi _i with variance σi _i is applied after the first linear transformations of the input parameter (one per neuron of the layer), wix+biw_i\,x+b_i, before the non-linear activation is applied: x→Linear Neuroni x→ Linear Neuron_i →(wix+bi)⋅Exp[−(wix+bi−μi)24σi2] →(w_i\,x+b_i)·Exp [- (w_i\,x+b_i- _i)^24 _i^2 ] =(wix+bi)⋅Exp[−(x+biwi−μiwi)24(σiwi)2] =(w_i\,x+b_i)·Exp [- (x+ b_iw_i- _iw_i )^24 ( _iw_i )^2 ] =(wix+bi)⋅Exp[−(x−μi′)24(σiwi)2] =(w_i\,x+b_i)·Exp [- (x- _i )^24 ( _iw_i )^2 ] (13) wherei=1,…,# neurons in layer 1. \ \ \ i=1,…,\# neurons in layer 1. The first line above is the transformation as shown in Bea:2024xgv , while the second and third lines are a mathematical rewriting that makes the GL effect more explicit, where the weight wiw_i is reabsorbed by dividing the bias bib_i and the mean μi _i together into a new variable μi′ _i . The values of the different μi′ _i can be chosen to be either fixed or learnable parameters. In the third line, one can see why this construction works. First, note that the response propagated through the network is only sensitive to the weights and biases that make wix+bi∼μi′w_i\,x+b_i _i . Thus, the N parameters wiw_i only affect the output solution in a neighbourhood of the input x∈μi′±σi/2x∈ _i ± _i/2. Furthermore, the fact that the factor σi _i is normalized by the weight wiw_i (while keeping σi _i fixed, as detailed below) solves a problem that arises if one naively applies a window Exp[(ϕ−μi)24σi2]Exp [ (φ- _i)^24 _i^2 ] with a learnable value of σi _i. In the naive case, a single neuron i can take over the whole first layer (e.g. σi→∞ _i→∞), and thus the localization is lost. However, by normalizing in the way presented above, the case where one neuron would take over the whole layer, namely σieff=σi/wi→∞σ^eff_i= _i/w_i→∞, implies wi≪1w_i 1 which makes the whole response in (13) close to zero. On the other hand, if the weight wiw_i is large, the region of x controlled by this weight is very narrow. This is a desirable outcome, as changes in wiw_i modify the N solution in regions that are very localized around x≈μix≈ _i. The fixed values of σi _i are taken to be of the order of (Δx)−1=(xi−xi−1)−1( x)^-1=(x_i-x_i-1)^-1 in the sampling of the input variable x, so that they cover the whole span of x. GL on the N-V (x=ϕx=φ): For the N-V network that reconstructs the inverse function V(ϕ)V(φ), the GL is applied to the single input variable ϕφ, following the approach of Bea:2024xgv . The scale parameters σi _i are fixed to values of order Δϕ−1 φ^-1, where Δϕ φ is defined based on a uniform discretization of the ϕφ, even though the actual ϕφ samples are generated by a separate neural network and are not uniformly spaced. The means μi _i are kept as trainable parameters, allowing the N-V sufficient flexibility to explore a wide class of configurations in order to recover V(ϕ)V(φ). GL on the N-Solver (x=Zx=Z): Additionally, in this work we also employ GL on the affine parameter variable Z, which enters as an input parameter to the N-Solver network. In this case, both μi _i and σi _i values are fixed to linearly spaced values between 0 and 11 (span of Z), and to (ΔZ)−1( Z)^-1, respectively. The intuition behind imposing GL on Z is the following: in the space of parameters, the N-Solver seems to have a better handle on the solution space when it is allowed to modify a solution ψi _i that corresponds to some BCs given by a point in the equation of state (Si,Ti)(S_i,T_i) while not changing the ones corresponding to points in the S(T)S(T) curve that are distant in the physics of the problem.222Note that different values of (S,T)(S,T) might be very close in the equation of state but may represent very different physics, thereby having very different values of the affine parameter Z as explained in Sec. 3.3.2. GL can help the N be more “flexible” in the learning process, improving a solution corresponding to (S,T,Z1)(S,\,T,\,Z_1) while barely modifying one that corresponds to (S,T,Z2)(S,\,T,\,Z_2), with Z1Z_1 and Z2Z_2 very different from each other; this occurs, for instance, with states near the false and true vacuum. 3.3.4 Additional losses Along with the standard loss function involving the squared residuals of the different ODEs that are to be simultaneously solved in this problem, here we will also make use of additional losses which enter the main loss function as defined in subsequent sections. As explained later in detail, these will have a notably smaller weight than the main residual ODE loss term, but will nevertheless play a very important role in guiding the NNs towards recovering the right functional form in each run. Moreover, each of these additional loss terms comprise physical information that is entirely contained in the input S(T)S(T) curve, such that they can simply be regarded as a valid exploit of the physics present when one is given an equation of state. Below, we introduce the two main additional loss terms that we adopted for the inversion problem a hand. UV/IR asymptotic form: In our current theoretical setup, the family of equations of state S(T)S(T) considered arises from a theory obtained by deforming a UV CFT by a relevant operator O, triggering an RG flow to an IR CFT. The entropy density interpolates between the thermodynamic behaviour characteristic of the UV CFT at high temperatures and that of the IR CFT at low temperatures. For a conformal theory deformed by a relevant operator, it can be shown that the relationship between the entropy and the temperature at high temperatures asymptotes to S=c1T3+c2T2Δ−5+⋯,S=c_1T^3+c_2T^2 -5+·s, (14) where c1,c2c_1,c_2 are numerical coefficients and Δ is the conformal dimension of O. In our previous work Bea:2024xgv , we already used this fact to extract information from the boundary data since at the UV the conformal dimension is set, ΔUV=3 _UV=3, and hence given any S(T)S(T) one can make a fit in the high -temperature regime to extract the coefficients c1UV,c2UVc^UV_1,c^UV_2. In our conventions, this gives333We note that the same equation appeared as Eq. (3.1) in Bea:2024xgv , where a typo caused the second term on the right-hand side to appear with a plus sign instead. S=π4T3−3π464T+⋯.S=π^4T^3- 3π^464T+·s. (15) By solving the EKG equations (5) perturbatively near the boundary, this form of the entropy density can be shown to fix the value of the scalar potential and its first two derivatives at the AdS critical point ϕ=0φ=0, which corresponds to the UV fixed point of the dual boundary theory: C0≡V(0)+3=0,C1≡V′(0)=0,C2≡V′(0)+3=0,C_0≡ V(0)+3=0, C_1≡ V (0)=0, C_2≡ V (0)+3=0, (16) defining the variables C0,C1,C2\C_0,C_1,C_2\ that were used to implement these values for the potential as an additional loss in the implementation of the N model. Having exploited the information at the UV from the given equation of state, we provided the PINNs model with these boundary conditions for the potential in the form of an additional loss. However, what was not attempted at the time was to similarly extract information at the IR from the equation of state; as it turns out, one can extend this strategy and play the same game by first making a fit at low temperatures to (14) to find c1IRc^IR_1 and the corresponding conformal dimension ΔIR _IR. Having previously found the value for c1UVc_1^UV shown in (15), one can use the relation (LIRLUV)3=c1IRc1UV ( L_IRL_UV )^3= c_1^IRc_1^UV (17) to find LIRL_IR given LUV=1L_UV=1.444Eq. (17) comes from the thermodynamic relation sT3=2π4L3κ52 sT^3= 2π^4L^3κ^2_5, where this relation is true when the theory is conformal at the UV/IR regimes. By equating said regimes, one can find the relation between the different AdS radii and the leading-order coefficient of the conformal theory deformed by the relevant operator. Then, we can use this value to find the theoretical value of the potential at the IR thanks to the relation C3≡V(ϕlast)=−3LIR2C_3≡ V( _last)=- 3L^2_IR (18) shown in attems2016thermodynamics ; Attems:2016tby , where the last value of the potential is at ϕ=ϕlastφ= _last. Since the IR is a fixed point, we know that the potential will also have a critical point at ϕ=ϕlastφ= _last, and so C4≡V′(ϕlast)=0.C_4≡ V ( _last)=0. (19) Lastly, the second derivative of the potential encodes the mass of the bulk scalar field at the IR fixed point via V′(ϕlast)=mIR2V ( _last)=m^2_IR. The standard AdS/CFT dictionary Witten:1998qj ; aharony2000large relates the mass of a scalar in AdSd+1AdS_d+1 to the conformal dimension of the dual operator via m2L2=Δ(Δ−d)m^2L^2= ( -d). Applied at the IR fixed point in our five-dimensional setup, from this we obtain C5≡V′(ϕlast)=ΔIR(ΔIR−4)LIR2,C_5≡ V ( _last)= _IR( _IR-4)L^2_IR, (20) as used in Attems:2016tby for this same family of holographic models.555Note that this is the IR-side analogue of the definition of C2C_2 in (16), which encodes the same relation at the UV fixed point with ΔUV=3 _UV=3 and LUVL_UV = 1. Once again, since these boundary conditions on the potential were implicitly contained in the given equation of state, we will make use of them in our N model. Note that these new additional losses will give information about the value of the potential at ϕlast _last, as well as its first and second derivatives there, but they contain no information on what this value ϕ=ϕlastφ= _last should be; the N model must still be able to learn this during training. We will employ these additional losses in both the near-false-vacuum and false vacuum regimes, where in the latter this will provide physical information on the global maximum and minimum points but not on the local extrema of the potential. Separating solutions among the two branches: For runs with equations of state in the false vacuum regime specifically, we will also introduce a new additional loss term. Firstly, as explained in previous sections, these potentials include a region from their local minimum point up until a certain numerical critical point whose values have no dual boundary thermal states, as in this region the dual thermal states belong to a different CFT at the boundary. Hence, the value of the scalar field at the horizon, ϕ(u=1)≡ϕHφ(u=1)≡ _H, will go from the maximum of the potential at ϕH=ϕ0=0 _H= _0=0 to the local minimum at ϕH=ϕFV _H= _FV and should then “jump” to some critical value ϕH=ϕc _H= _c such that boundary thermal states that come after the false vacuum state will have ϕH _H values larger or equal to this numerical critical point. To impose this physically motivated statement, we can effectively split the equation of state into two branches in order to separate thermal points before and after the false vacuum. The exact procedure for separating the branches will be explained in detail in the subsection below. For the first branch (before the false vacuum, ϕ∈[ϕ0,ϕFV]φ∈[ _0, _FV]), we do not impose any additional constraints on the loss besides the ODE residuals and the values of V and its first and second derivatives at the UV fixed point at ϕH=ϕ0 _H= _0 in the form of (16). The value of ϕH _H found by the N for the solution corresponding to the last point in the first branch of S(T)S(T) should, after an appropriate amount of training, approximate the position of the minimum of the potential corresponding to the false vacuum, ϕH=ϕFVNN _H= _FV^N. However, for the BCs coming from the second branch (after the false vacuum, ϕH>ϕFV _H> _FV), we will include an additional loss term that forces the corresponding solutions for ϕ(u)φ(u) to have a value at the horizon ϕH _H above the value ϕFV _FV found during the training of the first branch. In particular, this term can be expressed as: LFV=∑i∈ 2nd branchσ(ϕFVNN−ϕi),L_ FV= _i∈\,2^nd branchσ (φ^N_ FV- _i )\ \ , (21) where σ is the sigmoid function and ϕFVNNφ^N_FV corresponds to the position found by the N of the local minimum of the potential when trained only on points in the first branch, i.e. V(ϕFVNN)=min(VNN1(ϕ))V(φ^N_FV)=min (V N_1(φ) ). Note that this additional loss term will not be employed in runs outside the false vacuum regime, as there the potentials will not have the local extrema features and thus the procedure of splitting the input BCs into two branches will not be necessary. 3.4 N setup and procedure Having presented the improvements made on the original PINNs-based setup, we will now explain the procedures that will be used to approach the inverse problem in: first, the near-false-vacuum, and second, the false vacuum regimes, highlighting the differences in the implementation of the setup shown in Fig. 3. We will define the near-false-vacuum regime as the set of inverse problems lying in the range 0.5808<ϕM≤0.80.5808< _M≤ 0.8. The lower bound represents the potential that has a degenerate critical point, such that for a ϕM _M lower than it we enter the false vacuum regime, while the upper bound is the inverse problem with the most pronounced phase transition that was attempted in our previous work Bea:2024xgv . In this regime, the pipeline used will be the same that was employed in said study, but we will now adopt the improved architecture presented in the previous subsection along with its aforementioned novel features. More specifically, given some equation of state S(T)S(T): 1. The N-Solver network, parametrized by weights WDW_D, takes the following discrete inputs: • a collection of thermodynamic data points (Ti,Si)(T_i,S_i) satisfying the equation of state • a set of radial grid points un∈[0,1]u_n∈[0,1] representing the independent variable • a set of affine parameter values Zi∈[0,1]Z_i∈[0,1] indexing the thermodynamic data points along the curve For each pair (Ti,Si)(T_i,S_i) and each un,Ziu_n,Z_i values the network outputs predictions for the bulk fields A~(u,Z,(T,S)),Σ~(u,Z,(T,S)),ϕ(u,Z,(T,S)), A(u,Z,(T,S)), (u,Z,(T,S)), φ(u,Z,(T,S)), together with the auxiliary quantities νA(u,Z,(T,S)),νΣ(u,Z,(T,S)),νϕ(u,Z,(T,S)). _A(u,Z,(T,S)), _ (u,Z,(T,S)), _φ(u,Z,(T,S)). By construction, these functions exactly satisfy the boundary conditions (11). Each thermodynamic pair (Ti,Si)(T_i,S_i) specifies a complete bulk geometry. Derivatives with respect to u, namely νA′ _A , νΣ′ _ , and νϕ′ _φ , are obtained analytically through automatic differentiation. 2. The outputted scalar field values ϕi,n=ϕ(un,Zi,(Ti,Si)) _i,n=φ(u_n,Z_i,(T_i,S_i)) are then supplied to a second neural network, N-V, with parameters WVW_V. This network reconstructs the scalar potential by producing predictions for V(ϕi,n)V( _i,n). Its derivative with respect to ϕφ is again computed using automatic differentiation. 3. The outputs of both networks are inserted into the system of ODEs (10). A global loss function ℒL is defined and minimized with respect to all trainable parameters WDW_D and WVW_V via stochastic gradient descent. The loss function takes the form ℒ=∑α∑n∑iEα(un,Zi,(Ti,Si))2+λUV(C02+C12+C22)+λIR(1)(C3)2+λIR(2)(C4)2+λIR(3)(C5)2, splitL=& _α _n _i\ E_α (u_n,Z_i,(T_i,S_i) )^2\\ &+ _UV (C_0^2+C_1^2+C_2^2 )+ _IR^(1) (C_3 )^2+ _IR^(2) (C_4 )^2+ _IR^(3) (C_5 )^2, split (22) where the first term represents the squared residuals of all equations of motion (10), summed over the equation index α, over the radial grid points unu_n, and over all thermodynamic configurations (Ti,Si)(T_i,S_i) with their respective affine parameter ZiZ_i. The second term enforces the original additional loss controlling the three boundary conditions on the potential at the UV, (16). Finally, the three last terms represent the new additional losses related to the boundary conditions on the potential at the IR, with the terms C3,C4,C5C_3,C_4,C_5 defined above in subsection 3.3.4. In practice, these three conditions were separated into three additional losses as it was found through experimentation that it was beneficial for the model to have the freedom to learn each condition individually. The hyperparameters λUV,λIR(1),λIR(2),λIR(3) _UV, _IR^(1), _IR^(2), _IR^(3) control the strength of the additional losses. For λUV _UV, we begin the training with λUV=0 _UV=0 for the first 10610^6 epochs and subsequently set λUV=50 _UV=50 to impose the UV constraints more strongly as we continue the training. Similarly, we start with λIR(1),λIR(2),λIR(3)=0 _IR^(1), _IR^(2), _IR^(3)=0 and then set λIR(1)=10−2 _IR^(1)=10^-2 and λIR(2)=λIR(3)=10−6 _IR^(2)= _IR^(3)=10^-6 after 10210^2 epochs. These particular configuration choices, such as the values of the hyperparameters and the different stagings of the additional losses, were found through experimentation to be generally optimal for the training of the models. The optimization is performed using the Adam algorithm, which augments stochastic gradient descent with estimates of higher-order moments. 4. The above procedure is repeated over a fixed number of training cycles (epochs). At each epoch, the loss is evaluated to monitor convergence, and training proceeds until the residuals of all equations fall below a prescribed tolerance. Note that the order of the loss at which training will stop will be different for the two regimes explored in this paper; specific details are given in Section 4. The training of an N model inherently involves randomness, stemming both from the use of stochastic gradient descent as well as the random initialization of the learnable weights and bias parameters inside each neuron. Therefore, as done in our previous implementations of this PINNs-based methodology, when running the above pipeline we perform various different runs of order ∼106 10^6 epochs and then continue the training for the best run of the batch. For the false vacuum regime, we will also use the pipeline described above but will employ it following a different strategy that will allow us to overcome the challenges associated with recovering potentials in this regime. More specifically, we will adopt a two-step approach that will involve splitting the equation of state into two separate sections and feeding them sequentially to the N model. The first branch will span points going from high temperatures at the UV to the false vacuum point, while the second branch will go from the first point after the false vacuum to the true vacuum of the curve, forming an elliptical-shaped section as illustrated in Fig. 1(a). The criterion used to split the branches is that we associate points with high T and high S, together with points where the energy density is non-zero while the temperature goes to zero (see Fig. 1(b)), to the first branch. These points can also be identified from the S/T3S/T^3 plot (Fig. 1(c)), where the first branch points go to a constant as T→0T→ 0 that is larger than the constant c1IRc_1^IR from the IR fixed point. The rest of points are associated to the second branch. Having split the points in the equation of state into these two branches, we will firstly proceed to only give input data from the first branch to our PINNs setup. The pipeline followed will be the same as for the near-false-vacuum case presented above, with the notable difference that the loss function (22) will not include the additional loss enforcing the coefficients C3,C4,C5\C_3,C_4,C_5\ at the true vacuum, as this first branch does not contain points pertaining to said regime. In this initial training phase, the solutions and the potential are quickly obtained by the N model to a good accuracy level, since the points in this section of the equation of state represent states going from the UV to the IR akin to a scenario where one has a crossover transition (typically corresponding to values such as ϕM=5 _M=5), which is the transition that is most easily dealt with. From this run, a potential spanning [ϕ0,ϕFV][ _0, _FV] going from an initial maximum point to a local minimum is recovered, the latter corresponding to the recovered false vacuum thermal state at the boundary. Following this first training step, the weights in the N corresponding to the first section of the potential will be frozen (expressed as V1frozen(ϕ)V_1^frozen(φ)), and this frozen model will now be “attached” to a second (new) N, V2N(ϕ)V_2^N(φ). This second model is then fed the points in the S(T)S(T) pertaining to the second branch. As explained in Sec. 2.2, we know from the theory that these solutions pertaining to the second branch should have a value of the scalar field at the horizon of ϕFV<ϕc<ϕH _FV< _c< _H, and thus the recovered solutions ϕNφ^N should span the range [ϕc,ϕlast][ _c, _last]. Here, ϕlast _last corresponds to the global minimum of the potential (the true vacuum in the equation of state), while ϕc _c is the critical value corresponding to the first thermal state in the second branch after the false vacuum. Both values ϕc _c and ϕlast _last must be learned by the N-setup. The region between ϕFV<ϕH<ϕc _FV< _H< _c is the section of the potential that corresponds to thermal states that do not pertain to the boundary equation of state and whose potentials exhibit some of the exotic behaviour briefly discussed in Section 1, such as the so-called skipping flows. Recovering the presence of this gap from the ϕ(u)φ(u) solutions from each of the branches, along with the values of the potential evaluated within said gap (corresponding to the presence of skipping flows in the potential) is highly non-trivial, as the N only has information of the potential at the initial and final points of this region, which additionally contains a local maximum. The full potential is hence defined to be given by the first N if ϕH<ϕFV _H< _FV, and by the second one if ϕH>ϕFV _H> _FV through the sigmoid function σ, in a way that allows the argument above to be realized during training but that does not provide the N model with additional information besides that which can be obtained from the S(T)S(T) curve alone: V(ϕ)=V1frozen(ϕ)⋅σ(ϕFV−ϕH)+V2,renormNN(ϕ)⋅σ(ϕH−ϕFV),V(φ)=V frozen_1(φ)·σ( _FV- _H)+V^N_2,\, renorm(φ)·σ( _H- _FV), (23) where the renormalized potential output to the second, unfrozen N is: V2,renormNN(ϕ) V_2,\, renorm^N(φ) =V1frozen(ϕFV)+∑k=1n−11k!(ϕH−ϕFV)k⋅dkV1frozen(ϕ)dϕk|ϕH=ϕFV =V_1 frozen( _FV)+ _k=1^n-1 1k!\,( _H- _ FV)^k· . d^kV_1 frozen(φ)dφ^k |_ _H= _ FV +1n!(ϕH−ϕFV)n⋅V2N(ϕ) + 1n!\,( _H- _ FV)^n· V_2 N(φ) (24) This renormalization is such that the value of V2,renormNN(ϕH=ϕFV)V_2,\, renorm^N( _H= _FV) and the n−1n-1 first derivatives evaluated at ϕFV _ FV match with those of V1frozen(ϕ)V_1 frozen(φ). In our pipeline, we set n=4n=4. We consider the matching up to 3rd3^rd-order in derivatives of the potential to be enough for our purposes, since derivatives of the potential appear explicitly only up to second order in the Einstein field equations (EFEs) (10). For an explicit display of the matching-by-construction values of V and its derivatives at ϕH=ϕFV _H= _FV, see Fig. 4. Figure 4: Explicit display of the matching-by-construction of value of the two N-components of the potential V in (3.4) up to the 3rd3^rd order derivative. We further emphasize that ϕFV _FV is a quantity that has been obtained by the first N, from finding the value of the scalar field at the horizon, ϕH=ϕ(u=1) _H=φ(u=1), at the point with (S,T)≈(0,0)(S,T)≈(0,0) in the first branch of the equation of state S(T)S(T), and hence it does not imply additional information besides that which is obtained through running the described pipeline. In summary, the pipeline for the false vacuum cases will run with the same general details as the ones presented above for the near-false-vacuum regime, but will proceed in the following steps: 1. The N-Solver and N-V networks will undergo training using the architecture and activations shown in Fig. 3 on only S(T)S(T) points belonging to the first branch (UV to false vacuum). The loss function used will be: ℒ1=∑α∑n∑i∈ 1st branchEα(un,Zi,(Ti,Si))2+λUV(C02+C12+C22).L_1= _α _n _i∈\,1^st branchE_α (u_n,Z_i,(T_i,S_i) )^2+ _UV (C_0^2+C_1^2+C_2^2 ). (25) 2. Once trained to recover the potential from the initial maximum point to the local minimum, the weights of the N-V network are frozen. Then, a new N-Solver and new N-V network are initialized. The new N-Solver will give solutions with a new set of BCs. The new N-V is attached to the frozen N-V, and the full model is ran only on BCs given by points from the second branch, as shown in the architecture in Fig. 5. In this second stage of training, the loss function is modified to: ℒ2=∑α∑n∑i∈ 2nd branchEα(un,Zi,(Ti,Si))2+λIR(1)(C3)2+λIR(2)(C4)2+λIR(3)(C5)2+LFV, splitL_2= _α _n _i∈\,2^nd branch&\ E_α (u_n,Z_i,(T_i,S_i) )^2\\ &+ _IR^(1) (C_3 )^2+ _IR^(2) (C_4 )^2+ _IR^(3) (C_5 )^2\\ &+L_FV\ \ , split (26) including the additional losses at the IR global minimum of the potential ϕH=ϕlast _H= _last presented in subsection 3.3.4, and at ϕH=ϕFV _H= _FV shown in (21). Both values ϕlast _last and ϕFV _FV are learned by the N-setup. Figure 5: N architecture for the two-branch setup used on false vacuum equations of state. The N-V network, parametrized by weights WVW_V, is composed of two different models that are trained sequentially; firstly, the weights of N-V1V_1 are loaded from a pre-training carried out only on input data from the first branch. Then, the N-V2V_2 and N-Solver models are trained on the differential equations with only input data from the second branch, while keeping the weights in N-V1V_1 frozen throughout. The final output of both N-V models is combined to obtain the final potential V following expression (23). 4 Results Here, we present the results obtained by applying the pipeline presented in the previous section to the inversion problem at hand, aiming to recover the bulk scalar potential V(ϕ)V(φ) given an equation of state S(T)S(T) in the near-false-vacuum and false-vacuum regimes. 4.1 Near-false-vacuum regime (ϕM=0.8 _M=0.8) To make contact with our previous work Bea:2024xgv , we will first present results on the most difficult case attempted there, which being the ϕM=0.8 _M=0.8 case sits within the near-false-vacuum regime. For this problem, we will use the pipeline shown in Fig. 3 and described in detail in the previous section. The results for the potential obtained by the N model are shown in Fig. 6, where it is displayed against the theoretical potential together with the best result previously achieved in this case in our earlier work. The potential obtained from the N model in this near-false-vacuum regime, with only minor deviations from the theoretical potential, shows a significant improvement over the one previously obtained with the old N setup, displaying the strengths of the novel features adopted in our current pipeline. Moreover, to achieve this result the N model only required training for 2 million epochs, which is less than the 3.5 million epochs of training that the old models underwent to obtain a worse potential. The model was trained until the loss fell to an order of 10−610^-6, from which point further training was seen to lead to only little improvement. Figure 6: Inverted potential in the near-false-vacuum regime for ϕM=0.8 _M=0.8. We present results for the N-pipeline of this work (solid red) compared to the best result obtained in the same case from our previous work Bea:2024xgv (dash-dotted black) and the true theoretical function (dashed blue). (a) (b) Figure 7: Equation of state recovered from the inverted potential for ϕM=0.8 _M=0.8, presented as S(T)S(T) (a) and S/T3(T)S/T^3(T) (b). We show the results for this work’s N pipeline (solid red) compared to the input theoretical equation of state (dashed blue), and to the previous work’s results Bea:2024xgv for the same problem (dash-dotted black). (a) (b) Figure 8: Reconstruction errors for the near-false-vacuum regime with ϕM=0.8 _M=0.8. We show results for this work (solid green) and for the previous work Bea:2024xgv (dashed blue) for the same case. The rms value for each case is shown in the legend. (a): RE in the recovered potential VPINN(ϕ)V_PINN(φ) as a function of ϕφ, with a mean value of 1.2%1.2\%. (b): RE in the recovered equation of state TPINN()T_PINN(C), with a mean value of 3.3%3.3\%, where ≡S/T3C≡ S/T^3. In order to quantitatively evaluate the accuracy of the inverted potential, at first one might simply compare said potential from the N against the theoretical potential by calculating the relative error (RE) between the two. This comparison can be easily made once a potential is obtained from the N model by computing: δV(ϕ)=|Vtheory(ϕ)−VPINN(ϕ)Vtheory(ϕ)|.δ V(φ)= | V_theory(φ)-V_PINN(φ)V_theory(φ) |. (27) Additionally, we can compute the root mean square666The root mean square is defined as rms(%)=1n∑i=1nxi2⋅100%rms(\%)= 1n _i=1^nx_i^2· 100\%, where xix_i is in our case the RE. (rms) of the RE to obtain a single value that quantifies the error related to the recovered function. These results are shown in Fig. 8(a), where we can observe the RE to never exceed 4.7%4.7\% with an rms value of 1.84%1.84\%. Comparing these results with the ones presented in Bea:2024xgv for the same case of ϕM=0.8 _M=0.8 the old N-model obtained a maximum value for the RE of around 35%35\% and an rms of 48.34%48.34\% in its best run. Hence, we can observe a substantial improvement in terms of error magnitude, as the error in the recovered potential presented in this work shows an improvement of around 1 order of magnitude with respect to the previous best-case results for ϕM=0.8 _M=0.8 in Bea:2024xgv . Furthermore, the error quantifications obtained here are generally in agreement with the values obtained in earlier works for easier realizations of higher ϕM _M values, which is notable given the increased difficulty of the inversion problem in cases within the near-false-vacuum regime. Assessing the quality of the recovered potentials VPINN(ϕ)V_PINN(φ) as presented above is rigorous as a way of testing our N-pipeline. However, one could find the error in (27) unsatisfactory if viewed as a cyclical argument, since in principle one should be able to use the pipeline presented in this paper for an equation of state obtained in some other manner (for instance, from experimental sources) without requiring a prior theoretical holographic model, Vtheory(ϕ)V_theory(φ). Hence, for any equation of state used in our pipeline, Sinput(T)S_input(T), we will take the potential obtained from the N model, VPINN(ϕ)V_PINN(φ), and use it to solve the direct problem to recover an equation of state SPINN(T)S_PINN(T) which we can then compare with Sinput(T)S_input(T) to measure the accuracy of the model. Note, however, that this is a very stringent test, since the set of equations being solved both in the direct and inverse problems (namely EFE + KG) is highly non-linear, and thus small errors in the recovered potential VPINN(ϕ)V_PINN(φ) can be largely amplified when passing through these non-linear equations. The SPINN(T)S_PINN(T) curve obtained from carrying out the direct problem using the recovered N potential is shown in Fig. 7, where the curve SPINN/T3S_PINN/T^3 is additionally plotted to display the UV and IR behaviours more clearly. Despite the existent deviations from the true equations of state in the middle sections of the curves, which are attributed to the same small deviations that are present in the recovered potential in Fig. 6, the equations of state are generally in very good agreement with the theoretical input curves. Following the previous discussion, the RE in the recovered equations of state is calculated using: δT()=|Tinput()−TPINN()Tinput()|,δ T(C)= | T_input(C)-T_PINN(C)T_input(C) |, (28) where the function (T)≡S/T3C(T)≡ S/T^3. The RE in the recovered equation of state is calculated using T()T(C) instead of the usual S(T)S(T), T(S)T(S) or (T)C(T) due to the former not being multi-valued, unlike the latter three. The feature of multi-valuedness of T(S)T(S) is more pronounced in the false-vacuum regime (i.e. for ϕM=0.55 _M=0.55). These multi-valuedness arguments can be seen in Figs. 1(a) and 1(c). The resulting RE is shown in Fig. 8(b), where it can be seen to never exceed a value of around 6.1%6.1\% and has an rms value of 3.71%3.71\%. Comparing these values once again to those presented in Bea:2024xgv for the same case, there a maximum RE of around 62%62\% and an rms of around 15%15\% were obtained. Therefore, the results presented in this work represent improvements by approximate factors of 10 and 4 in the maximum RE and rms values, respectively, for the same quality tests performed. Hence, the RE for this recovered phase transition in the near-false-vacuum regime is comparable to that of the phase transitions recovered in Bea:2024xgv in simpler regimes. This is a notable result, as the number of additional difficulties related to the large separation of scales that come with recovering potentials in this regime made it inaccessible to the PINNs-based model up to now. 4.2 False vacuum regime (ϕM=0.55 _M=0.55) Figure 9: Recovered potential (solid red) compared to the theoretical potential (dashed blue) for ϕM=0.55 _M=0.55. The potential corresponding to the first branch (FB) ranges from the global maximum to the local minimum denoted as FV (False Vacuum), marked by blue and red dashed lines corresponding to its theoretical and N-recovered positions, respectively. The strong agreement in the two lines shows the potential for the FB is recovered to a high accuracy. The potential found for the thermal states in the second branch (SB) spans the region from the local to the global minimum, approximately tracing out the correct local maximum in between. The theoretical and N-recovered values for ϕc _c (dotted) and ϕmin _min (dash-dotted) are also shown. To explore the false vacuum regime, we chose to tackle an equation of state well within this domain, namely that corresponding to a dual ϕM=0.55 _M=0.55 theory. Although this choice implies that the dual potential will have a larger separation of scales between its characteristic local and global stationary points, complicating the inversion problem further, it also will have a more pronounced local maximum. Since this is a main feature in the potential which is not encoded by thermal states at the boundary data, it will prove useful for the N model to clearly distinguish it from the local minimum when recovering the potential. Through experimentation, we observed that if on the other hand one attempts the inversion task on theories closer to the false vacuum threshold, ϕM≃0.5808 _M 0.5808, the resulting local extrema are very close to being degenerate to a single inflection point, which is a highly fine-tuned feature that is very difficult for the model to accurately find. (a) (b) Figure 10: Equation of state recovered from the inverted potential for ϕM=0.55 _M=0.55. (a): S(T)S(T) for the first branch (FB) (solid red) and second branch (SB) (dash-dotted red), versus the theory (dashed blue). (b): S/T3(T)S/T^3(T) for the FB (solid red) and SB (dash-dotted red) versus the theory (dashed blue). The morphology of the recovered curve matches the inputted S(T)S(T), and the false and true vacuum points are correctly recovered. (a) (b) Figure 11: Reconstruction errors for the false vacuum regime with ϕM=0.55 _M=0.55. (a): RE in the recovered potential VPINN(ϕ)V_PINN(φ) as a function of ϕφ, with mean value 13.7%13.7\%. The dashed black line indicates the false vacuum point which divides the first and second branches represented in solid and dash-dotted green lines, respectively. (b): RE in the recovered equation of state TPINN()T_PINN(C), with mean value of 6.3%6.3\% in the FB (solid green) and 141.6%141.6\% in the SB (dash-dotted green), being ≡S/T3C≡ S/T^3. The FB curve has been shifted to lower values by Δ=60 =60 for visualization purposes. For this run, the N model was trained for 27 million epochs, at which point the loss fell to an order of 10−510^-5; further training from then was seen to lead to only little improvement. The results for the potential found by the N are shown in Fig. 9. First, it can be observed that the potential recovered from boundary data points corresponding to the first branch (FB), that is the section of the potential from its global maximum to its local minimum, is recovered to a very high accuracy, as shown by the strong matching between the theoretical and numerical positions of the local minimum (which corresponds to the false vacuum state), as well as the overall precise tracing of the theoretical potential. The RE in this region of the potential corresponding to the FB, namely [ϕ0,ϕFV][ _0, _FV], is below 1%1\%, as can be seen in Fig. 11(a). For boundary data from the second branch (SB), the N model is able to correctly find the local maximum feature as well as the global minimum point. The potential mostly deviates in the region whose dual thermal states are not part of the boundary data, which is to be expected, but remains remarkably close in morphology, content and general location of characteristic features to the true potential even with the considerable separation of scales present in the current theory. When it comes to the equation of state recovered using this learned potential, one can similarly see that it matches the original boundary curve to a good relative precision, including its crucial points near the origin corresponding to the false and true vacuum states. The pointwise REs for both the full learned potential and recovered S(T)S(T) are shown in Figs. 11(a) and 11(b), respectively. Here, it is important to note that the reported error values appear large precisely in the SB region without boundary input data not because of poor reconstruction, but as an artifact of normalization: both quantities are divided pointwise by a reference value (|Vth||V_th| and |Tinput||T_input|, respectively) that becomes small near the local extrema and the (S,T)≈(0,0)(S,T)≈(0,0) vacua, while the absolute deviations there are simultaneously largest owing to the absence of direct thermal data. The combination of small denominator and largest absolute error inflates the error ratio even where the recovered functions track the true ones closely. (a) (b) Figure 12: (a): Solutions for the scalar field ϕ(u)φ(u) for the case with ϕM=0.55 _M=0.55. The solutions below the dashed horizontal black line correspond to BCs from the FB of the S(T)S(T) curve (solid lines), while the ones above it correspond to points in the SB (dash-dotted lines). The N-model is able to recover the feature of the existence of a gap Δϕ φ in the values ϕ(u=1)=ϕHφ(u=1)= _H. This is a defining feature in false vacuum cases, where the region of values of the ϕH _H where ϕFV<ϕH<ϕc _FV< _H< _c corresponds to thermal states of a different dual theory. The labels “first”, “middle” and “last” indicate the correspondence of the BCs in the equation of state (panel (b)) with the obtained solutions. (b) Equation of state with BC points in the FB and SB corresponding to solutions of panel (a). With regards to the bulk metric field solutions obtained by solving the coupled ODE system, which the N model must do simultaneously in order to reconstruct the potential, evidence of the false vacuum can also be observed directly from the learned profiles of ϕ(u)φ(u) shown in Fig. 12. Looking at the figure, one can see the solutions from boundary input data coming from the FB only reach up to a certain maximum value of ϕ(u=1)=ϕHφ(u=1)= _H, represented by a dashed black line, captured by the solution in light blue whose dual thermal state is that of the false vacuum ϕFV _FV. These are the same values that were used in the additional loss defined in (21). Afterwards, there is a significant gap in the solution space quantified by Δϕ=0.5 φ=0.5 between this solution and the subsequent one for the ϕ(u)φ(u) dual to the first thermal state from the SB, whose value at the horizon is taken to define the quantity ϕc _c. This parameter is found by the N-model as a by-product of the training procedure, and for this run it was found to be ϕc∼1.32 _c 1.32. As explained above, this is an expected physical feature of the theory, since solutions with ϕFV<ϕH<ϕc _FV< _H< _c correspond to thermal states of a different QFT. Nevertheless, it is noteworthy that the N-model is able to identify this structure, as the thermal states whose dual ϕ(u)φ(u) solutions attain the delimiting horizon values ϕFV _FV and ϕc _c are parametrically very close in the equation of state and possess very similar (S,T)≈(0,0)(S,T)≈(0,0) values, despite corresponding to notably different physics in the bulk theory. It is remarkable that these acutely fine-tuned solutions and relevant physical features are recovered by the model when simultaneously solving for the EKG equations for different boundary conditions. We note that this is in part thanks to the introduction of the novel features presented in Section 3.3 such as the affine parameter, which helps break the degeneracy between the last point of the FB and first point of the SB (light blue triangle and green circle in Fig. 12(b), respectively). 5 Discussion In this work, we have presented a natural extension of the PINNs-based model of Bea:2024xgv applied successfully to new regimes previously inaccessible to N methodologies. The main achievements of this study are twofold. First, in the near-false-vacuum regime (ϕM=0.8 _M=0.8), the combination of the novel techniques introduced here, including the affine parameter for uniform curve sampling, Gaussian localization on the affine parameter direction, and the new additional losses enforcing boundary conditions at the IR fixed point yields a substantially improved reconstruction of the scalar potential compared to the results of Bea:2024xgv ; tarancon2025efficient , where this case represented the boundary of applicability of the PINNs-based methodology. The recovered equation of state, obtained by solving the direct problem with the N-reconstructed potential, agrees well with the input data. Secondly, in the false vacuum regime (ϕM=0.55 _M=0.55) we have successfully recovered the bulk potential for the first time from boundary thermodynamic data. This constitutes a qualitative advance over previous results, as the potential in this regime contains genuinely new features that are only indirectly encoded in the input equation of state. Specifically, solutions with ϕH _H in the region of the potential between the local minimum (the false vacuum) and a critical value ϕc _c do not correspond to thermal states of the original QFT whose thermodynamic curve S(T)S(T) is given as an input to the N. The fact that the model is nevertheless able to approximately reconstruct this “hidden” portion of the potential, including the correct identification of the local maximum, illustrates the constraining power of the Einstein equations when combined with boundary data from both sides of the gap. A related result concerns the solutions for the bulk scalar field ϕ(u)φ(u). As shown in the previous section, the N model is autonomously able to find the gap in the scalar field values at the horizon, ϕH _H, that separates solutions from the first and second branches. The thermal states whose dual solutions have ϕH _H values at the boundaries of this gap (ϕFV _FV and ϕc _c) are parametrically close in the (S,T)(S,T) plane, yet they correspond to very different bulk solutions. The ability of the model to resolve this near-degeneracy illustrates the usefulness of the affine parameter Z, which makes it possible for the N to clearly distinguish these distinct boundary conditions and obtain this important consistency check. Another notable feature used is the implementation of the two-branch training strategy employed for the false vacuum case. By first training on boundary data from the UV to the false vacuum, the model quickly and accurately recovers the section of the potential from the global maximum to the local minimum, a problem that is structurally similar to the crossover case in Bea:2024xgv . Freezing the weights of this model and then training a new N-Solver on the second branch allows the model to focus entirely on the more challenging portion of the potential, from ϕc _c to the global minimum. The smooth stitching of the two N-V models via a sigmoid transition and Taylor matching at the false vacuum point ensures continuity of the reconstructed potential and its derivatives. The generality of this decomposition strategy may prove useful more broadly for inverse problems in which the solution space has natural separation of scales or hierarchies. Several aspects of the current results invite further improvement. For the ϕM=0.55 _M=0.55 case, while the potential reconstruction in the “hidden” region between ϕFV _FV and ϕc _c captures the correct qualitative features, it equally shows quantitative deviations from the theoretical potential. This is to be expected: the N has no direct thermodynamic data in this region and must rely solely on the differential equations and the information available at the endpoints. It would be valuable to explore whether additional physical constraints could be incorporated as additional losses to help further constrain the potential. Another open direction concerns theories whose ϕM _M is close to (or is) the critical threshold ϕM≃0.5808 _M 0.5808, where the two local extrema are very nearly (or become) a degenerate critical point at which both V′V and V′V vanish. This regime is physically interesting because such theories provide models of out-of-equilibrium, self-sustained inflation Casalderrey-Solana:2025cdy . Through experimentation, we have observed that this is also a particularly challenging limit, as the fine-tuned structure in the small-scale features is difficult for the model to resolve. This is especially true when the potential exhibits a degenerate critical point, since in that case the N model must identify a feature consisting of a single point rather than an extended region of the potential. A systematic study of the model’s performance across a finer grid of ϕM _M values would help delineate the current boundaries of the method and identify where further technical advances are needed. Looking ahead, the results of this work open the door to several compelling applications, such as the study of false vacuum decay via bubble nucleation and the subsequent bubble dynamics in the holographic framework. Once a gravitational theory has been reconstructed for a theory of interest whose thermodynamics is known, the full machinery of holography becomes available for the study of far-from-equilibrium dynamics, including transport coefficients, thermalisation, and the approach to equilibrium. Finally, from a methodological perspective it is worth highlighting that the potential capabilities of the techniques developed and implemented in this work are not restricted to a holographic context. Instead, they address generic difficulties that arise when using PINNs to solve inverse problems involving large physical hierarchies, degenerate boundary conditions, and solution spaces with gaps or discontinuities. We expect them to be applicable to a broader class of inverse problems involving highly nonlinear differential equations. Acknowledgements.We acknowledge financial support from the “Center of Excellence Maria de Maeztu 2025–2029” award to the ICCUB, grant CEX2024-001451-M, funded by AEI/10.13039/501100011033. This work was also partially financed by a grant from the Simons Foundation (00017375, RJ). Partial funding for the work of RJ, PS, PTa and PTe was provided by project PID2022-141125NB-I00. The work of PS was supported by a grant from the “la Caixa” Foundation (ID 100010434), under the fellowship code LCF/BQ/DI23/11990074. PTa is supported by the project “Dark Energy and the Origin of the Universe” (PRE2022-102220), funded by MCIN/AEI/10.13039/501100011033. DM acknowledges financial support from Grant No. PID2022-136224NB-C22 from the Spanish Ministry of Science, Innovation and Universities, and from Grant No. 2021-SGR-872 funded by the Catalan Government. This research is also funded by the European Union (ERC, HoloGW, Grant Agreement No. 101141909). 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