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Janus: An Algorithm-Evaluator Co-Evolution Framework for LLM-Driven Discovery under Expensive Evaluation Budgets
Ximeng Liu, Qianlong Wang, Yingming Mao, Annan Li, Yatao Li, Shizhen Zhao, Jianmin Wu, Dawei Yin, Dou Shen
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
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Summary
The paper introduces Janus, a framework for LLM-driven program discovery in scientific domains where real evaluation is expensive. Janus co-evolves target programs and executable proxy evaluators generated by LLMs. It addresses label scarcity and distribution shift by maintaining region-conditioned portfolios of evaluators with online credit updates, using proxy predictions only to prioritize candidates for real validation.
Entities (11)
Relation Signals (10)
Janus → uses → LLM
confidence 95% · Janus uses LLMs to co-evolve target programs and executable proxy evaluators.
Proxy Evaluator → generatedby → LLM
confidence 92% · Janus leverages domain knowledge encoded in LLMs to generate task-specific evaluator programs
Janus → addresses → Label Scarcity
confidence 90% · To address label scarcity, Janus leverages domain knowledge encoded in LLMs.
Janus → addresses → Distribution Shift
confidence 90% · To mitigate distribution shift, Janus evolves evaluators alongside target programs.
Janus → evaluatedon → PyBaMM
confidence 90% · We evaluate Janus on five program-design benchmarks: PyBaMM...
Janus → evaluatedon → AQM
confidence 90% · We evaluate Janus on five program-design benchmarks: ... AQM ...
Janus → evaluatedon → FDTD Demux
confidence 90% · We evaluate Janus on five program-design benchmarks: ... FDTD Demux ...
Janus → evaluatedon → Reactor
confidence 90% · We evaluate Janus on five program-design benchmarks: ... Reactor ...
Janus → →
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Abstract
Abstract:LLM-driven program discovery relies on rapid evaluator feedback, but many scientific and engineering tasks require high-fidelity simulations, hardware execution, or physical experiments, making each evaluation expensive. Cheap surrogate evaluators can reduce this cost, yet fixed surrogates are vulnerable to search-induced distribution shift and are difficult to fit reliably from sparse, search-biased labels. We introduce Janus, a framework that uses LLMs to co-evolve target programs and executable proxy evaluators. To address label scarcity, Janus leverages domain knowledge encoded in LLMs to generate task-specific evaluator programs and calibrates them using real outcomes. To mitigate distribution shift, Janus evolves evaluators alongside target programs, selects them using a promotion-aligned objective, and maintains region-conditioned portfolios with online credit updates. Because proxy predictions remain fallible, Janus uses them only to prioritize candidates and requires real validation before candidates can enter the target-program population or update the incumbent. Across five scientific and engineering design tasks, Janus achieves a larger area under the best-so-far improvement curve over the real-evaluation budget and higher final performance than a matched baseline that evolves only target programs. On average, Janus reaches 99/% of the baseline's final improvement with 59.1/% fewer real evaluations. Evolved proxy evaluators also rank promising candidates more accurately than their seed versions. Together, these results extend evaluator-guided LLM discovery from tasks with cheap, scalable feedback to scientific domains where trustworthy evaluation is scarce and expensive.
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- Source: https://arxiv.org/abs/2608.08189v1
- Canonical: https://arxiv.org/abs/2608.08189v1
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Janus: An Algorithm–Evaluator Co-Evolution Framework for LLM-Driven Discovery under Expensive Evaluation Budgets Ximeng Liu123, Qianlong Wang3, Yingming Mao4, Annan Li3, Yatao Li5, Shizhen Zhao1, Jianmin Wu3, Dawei Yin3, Dou Shen3 Abstract LLM-driven program discovery relies on rapid evaluator feedback, but many scientific and engineering tasks require high-fidelity simulations, hardware execution, or physical experiments, making each evaluation expensive. Cheap surrogate evaluators can reduce this cost, yet fixed surrogates are vulnerable to search-induced distribution shift and are difficult to fit reliably from sparse, search-biased labels. We introduce Janus, a framework that uses LLMs to co-evolve target programs and executable proxy evaluators. To address label scarcity, Janus leverages domain knowledge encoded in LLMs to generate task-specific evaluator programs and calibrates them using real outcomes. To mitigate distribution shift, Janus evolves evaluators alongside target programs, selects them using a promotion-aligned objective, and maintains region-conditioned portfolios with online credit updates. Because proxy predictions remain fallible, Janus uses them only to prioritize candidates and requires real validation before candidates can enter the target-program population or update the incumbent. Across five scientific and engineering design tasks, Janus achieves a larger area under the best-so-far improvement curve over the real-evaluation budget and higher final performance than a matched baseline that evolves only target programs. On average, Janus reaches 99%99\% of the baseline’s final improvement with 59.1%59.1\% fewer real evaluations. Evolved proxy evaluators also rank promising candidates more accurately than their seed versions. Together, these results extend evaluator-guided LLM discovery from tasks with cheap, scalable feedback to scientific domains where trustworthy evaluation is scarce and expensive. I Introduction Large language models (LLMs) are increasingly being used to discover high-performing executable programs [39, 41, 11, 20]. For example, systems such as FunSearch and AlphaEvolve embed an LLM within an iterative discovery loop, in which the model proposes programs, an evaluator measures their quality, and the resulting feedback guides subsequent generations [27, 23, 6]. These systems have achieved remarkable results. Their success relies on the availability of an evaluator that can assess generated programs and provide timely feedback to guide the LLM’s subsequent proposals. However, many scientific and engineering tasks lack such timely feedback. In these settings, assessing a single candidate may require high-fidelity simulation, hardware execution, or physical experiments that take hours or even days. Evaluating every LLM-generated proposal is therefore impractical, making real evaluation, rather than candidate generation, the primary bottleneck. This mismatch between abundant candidate generation and scarce real-world feedback limits the applicability of existing LLM-driven search frameworks to scientific and engineering discovery. Figure 1: AlphaEvolve-style evaluator-guided discovery loop. The LLM proposes executable candidates; automated evaluators score them; the resulting successes and failures update the program database and condition later proposals. Figure 2: Evaluating a congestion-control design requires a suite of packet-level discrete-event simulations. A natural solution is to train a cheap surrogate using a small set of real evaluations and use it to screen candidates, as commonly done in Bayesian optimization, surrogate-assisted search, and model-based optimization [3, 8, 5, 34]. However, this approach faces two major challenges. First, distribution shift: because search is guided by surrogate predictions, the candidate distribution may move beyond the surrogate’s training support, causing prediction errors to accumulate and ultimately misdirect the search [40]. Second, label scarcity: real evaluations are limited and selected by the search process, making the available data both sparse and biased. In scientific problems governed by physical processes, different regions of the search space may exhibit qualitatively different behaviors as different physical mechanisms become active. With only sparse labels, a surrogate from a predefined model family may fail to identify these regime changes and generalize reliably beyond the observed data. To address these challenges, we introduce Janus, a framework that co-evolves target programs and executable proxy evaluators under a shared real-evaluation budget. To overcome label scarcity, Janus leverages domain knowledge encoded in LLMs to generate task-specific executable evaluator programs with plausible features, equations, and conditional structures, and calibrates them using limited real outcomes. To mitigate search-induced distribution shift, Janus evolves evaluators alongside target programs, selects them using a promotion-aligned objective, and maintains region-conditioned portfolios whose credits are updated online based on real evaluation results. Finally, to guard against residual proxy errors, Janus uses proxy predictions only to screen candidates and allocate the real-evaluation budget. Candidates remain provisional until they pass real validation, and the budget is allocated by balancing predicted quality, uncertainty, and novelty. We evaluate Janus on five program-design benchmarks: PyBaMM, AQM, FDTD Demux, Reactor, and Perishable IRP [26, 7, 36, 4, 35]. Across all five domains, Janus achieves a larger area under the best-so-far improvement curve over the real-evaluation budget and higher final performance than a matched baseline that evolves only target programs. On average, Janus reaches 99%99\% of the final improvement achieved by this baseline using 59.1%59.1\% fewer real evaluations. The evolved proxy evaluators also rank promising candidates more accurately than the initial versions. Together, these results extend evaluator-guided LLM discovery from tasks with cheap, scalable feedback to scientific domains where trustworthy evaluation is scarce and expensive. Problem Objective / artifact Validation procedure Validation time A. AlphaEvolve: final-result verification Fast matrix multiplication Rank-48 ⟨4,4,4⟩ 4,4,4 tensor decomposition Reconstruct the target tensor exactly and score the decomposition rank [23] ≈0.7≈ 0.7 ms 11-D kissing number 593-point integer configuration Check nonzero vectors and every pairwise separation constraint; score cardinality [23] ≈0.62≈ 0.62 s B. Deployment-scale validation in three Janus benchmark domains Battery fast charging (physical cells) Maximize cycle life under a 10-min charge Cycle a physical cell for 100 cycles (early label) or to failure [29, 2] ≈4≈ 4 d Network congestion control (ns-3 ECN) Minimize tail flow-completion time Simulate 5 s in ns-3 on a 384-rack, 6,144-host topology [42] 11–27 h Photonic cavity design (3D FDTD) Confine light in a compact cavity with minimal leakage Run one converged 3D MEEP FDTD solve (16 CPU cores, 32 GB) [22] ≈10≈ 10 h TABLE I: Evaluation-time comparison between two AlphaEvolve problems and representative scientific problems. I Motivating Observations I-A Ground Truth Can Be Expensive Evaluator-guided discovery is effective when candidate quality is cheap to verify. Figure 2 sketches this loop for AlphaEvolve. In fast matrix multiplication, algebraic identities certify correctness and multiplication count provides the score. This inexpensive feedback enabled AlphaEvolve to find a 4848-multiplication procedure for two 4×44× 4 complex matrices, improving on the 4949 multiplications required by two levels of Strassen’s algorithm [31, 23]. Many scientific tasks lack such cheap certificates and instead require high-fidelity simulation, hardware execution, or physical experiments. Figure 2 illustrates this problem in congestion control: evaluating one configuration requires packet-level ns-3 simulations across network settings, workloads, and random seeds, and can take hours [13, 42]. Table I shows the broader contrast: the AlphaEvolve examples take less than a second to verify, whereas representative scientific evaluations require hours or days. Evaluating every proposal is therefore impractical in these settings. I-B Fixed Proxies Are Not Enough A natural response is to train a cheap surrogate evaluator once and use it in place of the expensive ground truth. This idea is well established in Bayesian optimization, surrogate-assisted search, and offline model-based optimization: a learned model approximates an expensive objective, and the optimizer queries this model much more frequently than it queries the real evaluator [30, 3, 34]. However, a proxy may become inaccurate when evolution moves beyond its training distribution. It may then mistake a poor candidate for a good one and misguide the subsequent search. We illustrate this failure in protein engineering on the measured green fluorescent protein (GFP) fitness landscape [28]. A proxy trained near the wild-type sequence guides search over the broader sequence space. As selected candidates drift away from the training centroid, proxy scores rise to about 5.6 while measured fluorescence falls to about 2.9 (Figure 3), demonstrating search-induced exploitation of a frozen proxy. (a) Score misalignment. (b) Feature-space drift. Figure 3: Fixed-surrogate exploitation in GFP design. I-C Harnessing LLMs for Adaptive Proxies § I-B shows that a proxy must track the distribution induced by search. Yet refitting an arbitrary proxy online is insufficient: scarce high-fidelity labels leave each refresh with a small, search-biased archive. Scientific objectives can change qualitatively across different regions of the input space as different physical mechanisms become active. For example, a network can move abruptly from a low-loss regime to congestion once the offered load exceeds its service and buffering capacity. A conventional surrogate starts from a predefined model class, such as a GP, random forest, or MLP, and estimates its parameters only from labeled samples. It must discover such transitions from data alone, which is difficult with a small, search-biased archive. LLMs offer a way to introduce domain structure before fitting. We prompt an LLM with a description of the scientific problem and ask it to generate executable proxy functions. Drawing on its domain knowledge, the LLM can propose features, equations, and conditional branches that encode plausible physical mechanisms. Real labels are then used to select among the generated functions and calibrate their parameters. This narrows the search to physically plausible structures, making better use of scarce labels. To illustrate this advantage, we consider TCP NewReno incast in ns-3. Given a network setting, the task is to predict how the packet-drop rate changes with the number of concurrent senders and identify the congestion knee, defined as the first tested sender count at which the drop rate reaches 0.5%0.5\%. Every proxy is trained on the same 12 labeled simulations, with four sender counts measured under each of three training network settings, and is tested on eight unseen settings. We compare GP, random-forest, MLP, quadratic-ridge, and RBF-SVR proxies with an LLM-generated arithmetic proxy [30, 10]. Figure 4(a) shows that the LLM proxy most closely follows the measured drop-rate curve and captures the transition at N=16N=16 in one training setting. Figure 4(b) shows that it also has the lowest mean knee-location error across the eight unseen settings: 0.5000.500 tested sender-count positions, compared with 2.1252.125 for the strongest conventional baseline. The LLM can draw on networking knowledge to encode the capacity-driven transition and post-congestion behavior before sparse labels are used for calibration. (a) Diagnostic context A. (b) Knee error (grid steps). Figure 4: Few-shot TCP-incast knee recovery. Together, the insights from § I-B and § I-C motivate Janus. To extend evaluator-guided discovery to scientific domains where real evaluation is expensive, Janus uses an LLM to generate and update proxy evaluators online from the task description and accumulated real labels. Janus therefore co-evolves not only target programs but also the proxy evaluators that guide the search. I Janus Design I-A Overview Problem statement. We study program discovery for scientific tasks, where candidate programs must be evaluated by an expensive real evaluator. Let X denote the program space, and let freal(x)f_real(x) return a utility y(x)∈ℝy(x) and a validity label v(x)∈0,1v(x)∈\0,1\ for each candidate x∈x . Given a budget of B real-evaluation attempts, the objective is to find the valid program with the highest observed real utility: xB⋆∈argmax(x,y,v)∈ℋB:v=1y,x_B ∈ _(x,y,v) _B:\,v=1y, (1) where ℋBH_B denotes the real-evaluation outcomes. Framework components. Janus builds on AlphaEvolve’s island-based LLM program-search framework [23]. Each island i maintains a real-validated algorithm population PalgiP_alg^i, while the LLM generates offspring from selected parents. Janus additionally maintains an evaluator population PevalP_eval, a real-outcome archive A for fitting and selecting evaluators, and an island-local pool iC_i of proxy-scored candidates awaiting real evaluation. An active portfolio Π , selected from PevalP_eval, screens and prioritizes candidates. Framework pipeline. As shown in Algorithm 1, Janus interleaves target proposal, real evaluation, evaluator evolution, and archive refresh. LLM-generated target programs are provisionally scored by the active evaluator portfolio; selected candidates are then real-evaluated to update ℋH, A, and the valid target population. In parallel, LLM-generated evaluators are fitted and selected on A, while each archive refresh updates Π and re-scores provisional candidates. To make this pipeline work, Janus represents proxy evaluators as LLM-generated programs (§ I-B), selects evaluator portfolios for target-program clusters using a promotion-aligned meta-objective and updates their credit from online real outcomes (§ I-C), and prevents proxy errors from contaminating the target population through exploration-aware, real-validated promotion (§ I-D). Algorithm 1 Janus Pipeline 0: LLM; frealf_real; budget B; promotion period τp _p; evaluator-evolution period τe _e; refresh batch size b 1: Initialize all state and attempt count a 2: while a<Ba<B do 3: Choose island i with local step tit_i 4: if timodτp=0t_i _p=0 then 5: Real evaluation: choose x from iC_i or generate it directly 6: Evaluate x with frealf_real; update ℋH, A, and a 7: If v(x)=1v(x)=1, update PalgiP_alg^i 8: else if timodτe=0t_i _e=0 and |||A| is sufficient then 9: Evaluator evolution: generate e by LLM mutation from PevalP_eval 10: Fit and meta-evaluate e on A; update PevalP_eval and Π 11: else if Π≠∅ ≠ then 12: Target proposal: generate x by LLM mutation from PalgiP_alg^i 13: Score x with the region-conditioned subset Πi(x)⊆Π _i(x) 14: Enqueue x provisionally in iC_i 15: else 16: Cold start: generate x from PalgiP_alg^i and evaluate it with frealf_real 17: Update ℋH, A, a, and, if valid, PalgiP_alg^i 18: end if 19: if b new real outcomes have entered A since the last refresh then 20: Archive refresh: refit evaluators, update Π , and re-score candidates 21: end if 22: end while 23: return argmax(x,y,v)∈ℋ:v=1y _(x,y,v) :\,v=1y I-B Evaluator-as-Program Evaluator representation. Rather than fixing a surrogate family in advance, Janus represents each proxy evaluator as an executable program: e=(gϕ,θϕ()),e= (g_φ, _φ(A) ), (2) where gϕg_φ is the program structure generated by an LLM, and θϕ() _φ(A) contains the values of any adjustable parameters in that program after they have been fitted using the real-outcome archive A. For example, these parameters may include coefficients in a generated equation, thresholds in conditional branches, or parameters used to calibrate the program’s outputs. The dependence on A indicates that these values are updated as new real outcomes enter the archive. Separating the generated structure from its fitted parameters allows Janus to adapt an existing evaluator to new observations without asking the LLM to regenerate its code. If a generated program has no adjustable parameters, θϕ() _φ(A) is empty. The framework requires only a common interface for producing a candidate-selection score; within it, the LLM may construct features, equations, conditional branches, low-fidelity computations, or learned corrections. LLM-based evaluator generation. At each evaluator-evolution event, Janus selects a parent and structurally diverse inspirations from the clustered evaluator population PevalP_eval. The LLM receives the scientific task and real-evaluation specifications, the selected evaluator programs, and archive-derived diagnostics and judge feedback. It then outputs an executable Python evaluator implementing the required interface. The framework checks the generated artifact, executes it under a timeout, fits θϕ() _φ(A), and evaluates it on A using the promotion-aligned objective introduced in § I-C. Evaluators that pass these checks can enter PevalP_eval, and their observed ranking errors and judge feedback are included in subsequent LLM prompts. Thus, the LLM proposes the proxy structure, while real outcomes calibrate its parameters and determine whether it is retained. I-C Evaluator Selection and Adaptation Janus first maintains the real-outcome data used to fit proxy evaluators, then fits and selects evaluators according to their ability to identify promising candidates, and finally adjusts their use based on online performance. Real-outcome archive. The archive A is bounded, deduplicated, and balanced rather than maintained as a first-in, first-out history. It retains high-utility valid programs to preserve the real frontier, recent programs to reflect the current search distribution, valid–invalid boundary cases to expose feasibility errors, and structurally diverse programs to avoid duplicate-heavy evaluator assessment. After a batch of new real labels arrives, Janus advances the archive revision, refits and re-evaluates the leading evaluators, reselects Π , and re-scores stale provisional candidates. Evaluator fitting and selection. Ultimately, the proxy evaluator serves to help search discover better programs. Accurately identifying the highest-utility candidates therefore matters more than minimizing average prediction error across the archive. Janus accordingly scores each fitted evaluator by how well its top ranking recovers the best programs in A: Fmeta(e)=λR(1−Rk(e))+λNNk(e)+λρρ(e),F_meta(e)= _R (1-R_k(e) )+ _NN_k(e)+ _ρ(e), (3) where Rk(e)R_k(e) is the normalized regret, indicating whether the evaluator’s predicted top-k contains a near-optimal program, Nk(e)N_k(e) is the normalized discounted cumulative gain (NDCG), indicating how closely the predicted top-k matches the true top-k ranking, and ρ(e)ρ(e) measures global ranking consistency across the archive. The first two terms receive most of the weight because they directly measure the quality of candidates sent to real evaluation, while the correlation term provides a weaker global signal. Portfolio and online credit. Different evaluators may be reliable in different regions of the evolving program distribution. For each island, Janus treats the clusters in the current target-program population as regions and represents each region by the centroid of its programs’ code embeddings. A new candidate x is assigned to the nearest region according to cosine distance: r(x)=argminrdcos(z(x),μr),r(x)= _rd_cos (z(x), _r ), (4) where r ranges over the target-program regions on island i, z(x)z(x) is the candidate’s code embedding, μr _r is the centroid of region r, and dcosd_cos denotes cosine distance. After determining r(x)r(x), Janus selects the K evaluators in the active pool Π with the highest current credit in that region, forming the region-specific portfolio Πi(x) _i(x) that jointly scores x. The meta-evaluation scores used to initialize these credits are refreshed only after a new batch of real labels is incorporated into A. Between batch refreshes, whenever a promoted candidate receives a real outcome, Janus immediately updates the credit of every evaluator in Πi(x) _i(x) that supported its promotion: Fe,r=(1−αe,r)Fmeta(e)+αe,rge,r,F_e,r=(1- _e,r)F_meta(e)+ _e,rg_e,r, (5) Here, e∈Πe∈ indexes an active evaluator, r indexes an island-local target-program region, and Fe,rF_e,r is the current credit of evaluator e in region r. Fmeta(e)F_meta(e) is its batch-level score from Eq. (3), ne,rn_e,r is the number of candidates it helped promote in r that have received real evaluations, and κ>0κ>0 controls how quickly this evidence overrides the batch-level score. Thus, αe,r=ne,r/(ne,r+κ) _e,r=n_e,r/(n_e,r+κ) weights the regional evidence, while ge,rg_e,r combines the fraction of these candidates that are valid and improve the incumbent with their normalized mean improvement. Valid improvements therefore increase an evaluator’s regional credit, whereas invalid or non-improving recommendations reduce its weight for subsequent candidates in the same region. Portfolio prediction. For a candidate x, Janus obtains three quantities from each evaluator e∈Πi(x)e∈ _i(x): a normalized score s^e(x)∈[0,1] s_e(x)∈[0,1], a predicted validity probability pv,e(x)p_v,e(x), and an uncertainty estimate σe _e. The evaluator produces the first two, while Janus estimates σe _e as the standard deviation of its normalized score errors on the real-valid archive entries. Janus normalizes the region-specific credits Fe,r(x)F_e,r(x) into weights and uses the weighted averages of s^e(x) s_e(x) and pv,e(x)p_v,e(x) as the portfolio score s¯(x) s(x) and validity probability p¯v(x) p_v(x). The portfolio uncertainty σ¯(x) σ(x) combines the weighted evaluator uncertainties with the weighted standard deviation of their scores on x, thereby capturing both historical prediction error and disagreement within the portfolio. Domain Evaluator Design goal Battery PyBaMM Fast charge; cycle life Network ns-3 AQM Throughput; delay/loss FDTD Demux 2D FDTD Low-crosstalk routing Reactor Stiff ODE Yield; thermal safety Perishable IRP Routing sim. Low shortage/waste TABLE I: Cases and real evaluators. (a) PyBaMM (b) AQM (c) FDTD Demux (d) Reactor (e) Perishable IRP Figure 5: Results across five program-design domains under matched real-evaluation budgets. I-D Real-Anchored Promotion Provisional screening. Even after selection and online adaptation, proxy evaluators remain fallible under distribution shift, calibration error, or incorrect output scales. Janus therefore uses their predictions only to allocate the real-evaluation budget. Between real-evaluation events, the LLM generates offspring from real-validated parents, and the portfolio defined in § I-C produces (s¯(x),p¯v(x),σ¯(x))( s(x), p_v(x), σ(x)) for each offspring. Candidates with usable predictions are stored in a bounded and deduplicated pool iC_i and are rescored when the archive or evaluator portfolio changes. Budget allocation and validation. At each promotion event, Janus ranks candidates that pass basic predicted-validity and quality gates using α(x)=wqp¯v(x)s¯(x)+wup¯v(x)σ¯(x)+wnη(x),α(x)=w_q p_v(x) s(x)+w_u p_v(x) σ(x)+w_nη(x), (6) where α(x)α(x) is the acquisition score; s¯(x) s(x), p¯v(x) p_v(x), and σ¯(x) σ(x) are the portfolio’s predicted quality, validity probability, and uncertainty; and wq,wu,wn≥0w_q,w_u,w_n≥ 0 are their respective weights. Here, η(x)η(x) measures embedding-region novelty based on the cosine distance from the code embedding of x to its nearest target-region centroid; candidates farther from existing regions are considered more novel. The first term drives exploitation by favoring candidates expected to be both valid and high-quality. The second explores uncertain but plausibly valid candidates; multiplying both terms by p¯v(x) p_v(x) avoids spending budget on candidates predicted to be invalid. The final term explores underrepresented regions independently of the current validity prediction. After promotion, Janus discards the proxy predictions and calls frealf_real; only a real-valid outcome may enter PalgiP_alg^i or update the incumbent. Every outcome updates A and the online credit of its recommending evaluators. Because acquisition alone could suppress candidates misjudged by the entire portfolio, Janus also periodically sends unranked LLM offspring directly to frealf_real. IV Experiments IV-A Experimental Setup Benchmarks. We evaluate five program-design benchmarks (Table 5): PyBaMM (battery fast charging) [26], AQM (ns-3 active queue management) [7], FDTD Demux (two-dimensional FDTD wavelength demultiplexing) [36], Reactor (constrained exothermic reactor control) [4], and Perishable IRP (stochastic perishable inventory routing) [35]. Appendix A provides objectives, simulation settings, and invalidity checks. Comparison baselines. We compare Janus with (1) Alg.-Only, a matched algorithm-search baseline that uses the same task-program seed and real-evaluation budget but does not evolve evaluator programs, following the AlphaEvolve-style mechanism of evolving target programs against a fixed evaluator [23]; (2) Direct-LLM, which scores each candidate by prompting the LLM with the labeled history, following the regression-only LAEA setting [12]; (3) Embed-GP, which fits a Gaussian-process surrogate on fixed code embeddings of the real-evaluated candidates and uses its predictions to select candidates for real evaluation [30]; and (4) Human Baselines: CLO, D-RED, CMT, OCP, and an order-up-to heuristic for PyBaMM, AQM, FDTD Demux, Reactor, and Perishable IRP, respectively [2, 9, 36, 4, 24, 35]. Metrics. We report (1) AUBCΔ, the average best-so-far improvement over the real-evaluation budget; (2) BestΔ@B, the terminal improvement; and (3) Calls@99%, the real-evaluator calls needed to reach 99%99\% of the Alg.-Only terminal improvement. For evaluator adaptation, NDCG@3 measures top-three ranking quality, while normalized best regret@3 measures whether the predicted top three recover a near-optimal candidate. AQM (B=200B=200) FDTD Demux (B=70B=70) ID Change AUBCΔ (%) ↑ BestΔ@B (%) ↑ AUBCΔ (%) ↑ BestΔ@B (%) ↑ F Full Janus 5.5±0.45.5± 0.4 6.0±0.16.0± 0.1 316.3±6.9316.3± 6.9 655.7±119.5655.7± 119.5 S No evaluator evolution 4.1±1.0(↓25.5%)4.1\!±\!1.0\;( 25.5\%) 5.2±0.7(↓13.3%)5.2\!±\!0.7\;( 13.3\%) 300.4±45.0(↓5.0%)300.4\!±\!45.0\;( 5.0\%) 429.9±77.4(↓34.4%)429.9\!±\!77.4\;( 34.4\%) A1 One evaluator only 4.3±1.6(↓21.8%)4.3\!±\!1.6\;( 21.8\%) 5.1±1.2(↓15.0%)5.1\!±\!1.2\;( 15.0\%) 243.9±15.5(↓22.9%)243.9\!±\!15.5\;( 22.9\%) 340.7±61.7(↓48.0%)340.7\!±\!61.7\;( 48.0\%) M0 Global Spearman objective 4.8±0.9(↓12.7%)4.8\!±\!0.9\;( 12.7\%) 5.8±0.1(↓3.3%)5.8\!±\!0.1\;( 3.3\%) 252.0±31.0(↓20.3%)252.0\!±\!31.0\;( 20.3\%) 404.7±6.7(↓38.3%)404.7\!±\!6.7\;( 38.3\%) O0 No online credit updates 4.4±0.2(↓20.0%)4.4\!±\!0.2\;( 20.0\%) 4.9±0.1(↓18.3%)4.9\!±\!0.1\;( 18.3\%) 245.1±21.6(↓22.5%)245.1\!±\!21.6\;( 22.5\%) 382.5±101.5(↓41.7%)382.5\!±\!101.5\;( 41.7\%) TABLE I: Component ablations on AQM and FDTD Demux. IV-B Main Results For each method-domain pair, we run three random seeds. Each curve shows mean best-so-far improvement, with shading spanning the minimum and maximum across runs. Consistent budget efficiency. Janus achieves both a larger AUBCΔ and a higher terminal mean than Alg.-Only in all five domains. The AUBCΔ gap in budget-averaged improvement ranges from 1.61.6 percentage points on AQM to 45.145.1 points on Reactor, indicating that Janus performs better throughout the search, not only at the endpoint. Janus also finishes above the strongest human-designed reference shown in every panel, whereas Alg.-Only remains below the CMT reference on FDTD Demux and the OCP reference on Reactor. Calls to the Alg.-Only endpoint. To compare sample efficiency, we set the target in each domain to 99%99\% of the final improvement achieved by Alg.-Only and count how many real-evaluator calls each method needs to reach it. Janus needs 3232 versus 8282 real-evaluator calls on PyBaMM, 100100 versus 222222 on AQM, 4040 versus 4848 on FDTD Demux, 33 versus 1515 on Reactor, and 55 versus 2929 on Perishable IRP. Thus, Janus reaches the target earlier in all five domains, reducing the required real-evaluator calls by 16.7%16.7\%–82.8%82.8\% across cases, with a mean reduction of 59.1%59.1\%. Alternative proxy baselines. (1) Embed-GP. Embed-GP assumes that distances between fixed code embeddings reflect real utility, although semantic similarity may not preserve the task-specific numerical or physical behaviors governing performance. It underperforms Alg.-Only in both AUBCΔ and final improvement in four of the five domains: PyBaMM, AQM, FDTD Demux, and Perishable IRP. (2) Direct-LLM. Direct-LLM scores each candidate using a growing, heterogeneous history of up to 50 evaluated programs. Such long prompts may dilute relevant evidence, and the method does not explicitly model uncertainty or distribution shift. It therefore achieves lower AUBCΔ than Janus in all five domains, particularly on AQM and FDTD Demux, while consuming on average 8.1×8.1× as many LLM tokens per completed real-evaluator call as Janus (see Appendix E). IV-C Evolving Better Evaluators Ranking Gains from Evaluator Evolution. We test whether evaluator evolution improves the ranking decisions used to select candidates for real evaluation. For each run, we refit the seed evaluator and the highest-weight evolved evaluator on the same checkpoint archive, then compare their NDCG@3 on the same 40 later candidates excluded from fitting. Figure 6 shows that NDCG@3 increases from 0.6880.688 to 0.9700.970 on PyBaMM, from 0.0000.000 to 0.6940.694 on AQM, and from 0.6120.612 to 0.9230.923 on FDTD Demux, giving gains of 0.2820.282, 0.6940.694, and 0.3110.311, respectively. Appendix F examines the resulting program changes. For example, the evolved PyBaMM evaluator adds peak temperature, SEI growth, voltage headroom, and high-SOC duration, better reflecting the constraints enforced by the real evaluator. Zero-shot reuse under objective changes. When the objective changes modestly, an existing evaluator can be reused with a simple adjustment, without new labels or refitting. For example, on the reweighted FDTD Demux objective, the previously evolved evaluator continues to rank candidates accurately, achieving an NDCG@3 of 0.959 (see Appendix C). Figure 6: Ranking-accuracy gains from evaluator evolution. IV-D Ablation Study We ablate evaluator evolution (S), replace the evaluator portfolio with one evaluator (A1), replace the promotion-aligned meta-objective with global rank correlation (M0), and disable online credit updates (O0) on AQM and FDTD Demux. Table I reports AUBCΔ and terminal improvement over the shared seed under matched real-evaluation budgets, with full Janus (F) as the reference. All four ablations reduce both metrics in both domains, showing that the four components address complementary errors. On AQM, burst traffic distinguishes controllers with similar average utilization; evolved evaluators capture these differences through probe-response and burst-specific signals, while limited headroom makes false-positive promotions especially costly. This explains AQM’s sensitivity to evaluator evolution and online credit updates. In FDTD Demux, coarse-to-fine simulation error varies across layout morphologies, favoring a region-weighted portfolio over a single evaluator. Additional experiments in Appendix D examine the effect of the LLM backbone. Across the tested backbones, Janus consistently achieves a higher terminal improvement. V Related Work LLM-driven algorithm discovery. FunSearch and AlphaEvolve place LLM generation inside evaluator-guided population search, while Evolution of Heuristics and MCTS-AHD improve heuristic generation and search organization [27, 23, 19, 43]. These methods evolve target programs while assuming that the task evaluator is externally specified. Janus instead makes the executable proxy evaluator part of the search. Surrogate-assisted optimization. Bayesian optimization and surrogate-assisted evolutionary algorithms reduce expensive objective calls by fitting cheap response models and selecting which candidates receive real evaluations [30, 3, 8, 5]. Multi-fidelity methods additionally exploit approximations at different costs or resolutions [14]. Even adaptive approaches generally fit, evolve, or select models within predefined families or component libraries [25, 10, 1, 15]. Janus instead uses an LLM to generate task-aware executable proxy structures from domain descriptions and real outcomes. LLM-assisted proxies and co-evolution. LAEA uses an LLM directly to predict the quality of numerical candidates, while CoE-SAEA and LLM-SAEA select predefined surrogate models and candidate-selection rules [12, 38, 37]. LAEA’s numerical reproduction pipeline cannot be transferred unchanged to free-form programs; our Direct-LLM (LAEA-style) diagnostic retains its core mechanism of predicting a new candidate directly from historical real outcomes. These methods do not generate task-specific executable proxies that evolve with the search distribution. Related methods adapt rewards, opponents, instances, or evaluators rather than approximate a fixed, expensive scientific objective [21, 18, 17, 16, 44]. Janus instead co-evolves executable proxies while real outcomes remain the source of target fitness. VI Conclusion We introduced Janus, which uses LLMs to co-evolve target programs and executable proxy evaluators under limited real-evaluation budgets. 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The target program implements solve(problem) and returns charge stages, each specifying a current and an upper state-of-charge breakpoint. The evaluator runs a PyBaMM single-particle model with lumped thermal dynamics, SEI growth, and irreversible lithium plating [32]. It simulates 30 cycles at each of three ambient temperatures (1515, 2525, and 35∘35\, C), including a reference discharge for capacity measurement. A candidate is invalid if it fails to reach 60%60\% state of charge within 600 seconds, exceeds 65∘65\, C, crosses the plating-overpotential limit, violates the protocol bounds, or fails numerically. For a valid protocol, an early-life capacity-fade predictor [29] estimates the end-of-life cycle count; the score is the three-temperature mean divided by 2,000. ns-3 active queue management. The problem is to design an active queue management controller that maintains high bottleneck utilization while limiting queueing delay, loss, and flow imbalance [7, 33]. The target program implements control(obs): it observes queue length, average delay, recent drops and ECN marks, recent utilization, and the number of active flows, then returns RED/ECN thresholds, marking probability, queue limit, and an ECN switch. A lightweight queue model calls the controller every 0.250.25 seconds to generate an action schedule, which is replayed in an ns-3 dumbbell simulation [13]. The evaluator averages six 8-second packet-level runs: long-flow, burst-dominated, and mixed traffic, each under two fixed seeds, over a 20-Mb/s bottleneck. For each run, it computes g−d95−2ℓ−0.5(1−f)g-d_95-2 -0.5(1-f), where g is normalized goodput, d95d_95 is the 95th-percentile queueing delay in seconds, ℓ is loss rate, and f is Jain fairness; malformed actions or simulator failures invalidate the candidate. FDTD Demux. The problem is to design a dielectric layout that routes two wavelengths from a shared input waveguide to different output ports [36]. The target program implements design(nx, ny, spec) and returns a resolution-parametric relative-permittivity pattern for the rectangular region between one input and two output waveguides. The evaluator uses a fine-grid, two-dimensional TM-mode FDTD simulation at both wavelengths. If TiXT_iX denotes normalized power from wavelength i reaching output X, the score is 0.5(T1A+T2B)−0.25(T1B+T2A)0.5(T_1A+T_2B)-0.25(T_1B+T_2A): it rewards routing the first wavelength to port A and the second to port B while penalizing crosstalk. The default real grid is 480×320480× 320 cells with 10,000 time steps per wavelength. An incorrect array shape, non-finite pattern or field, or failed simulation invalidates the candidate. Exothermic semi-batch reactor. The problem is to control a semi-batch reactor to maximize desired product formation while avoiding thermal runaway and undesired side products [4]. Reactant B is fed into an initial charge of A; the desired exothermic reaction A+B→CA+B→ C competes with the temperature-accelerated side reaction C→DC→ D. The target program implements design(problem) and returns ten piecewise-constant feed rates and ten jacket temperatures for a one-hour batch. The evaluator integrates the full stiff six-state material and energy balance under tight numerical tolerances and averages (nC−nD)/85(n_C-n_D)/85 over three operating scenarios with different cooling capacities and runaway margins. Exceeding the scenario-specific temperature limit or the 85-mol feed budget, producing a malformed schedule or nonphysical state, or failing the solver invalidates the candidate. Perishable IRP. The problem is to choose daily replenishment quantities for perishable inventory under uncertain demand while accounting for purchase, routing, stockout, holding, and spoilage costs [35]. The target program implements policy(state) and returns next-day shipment quantities for eight retailers from age-bucket inventory, three-day forecasts, recent demand, service levels, capacities, locations, and costs. The evaluator enforces vehicle and store capacities, ages four-day-life inventory in FIFO order, realizes candidate-independent demand, and constructs a canonical nearest-neighbor delivery route. It runs 36 fixed independent 180-day episodes. Each episode receives 1−C/(12D)1-C/(12D), where C is total purchase, holding, waste, lost-sales, vehicle, and distance cost and D is total demand; the final score is the episode mean minus 0.150.15 times its standard deviation. Invalid shipment vectors or capacity violations invalidate the policy. Appendix B Direct-LLM Baseline Protocol Program-space adaptation of LAEA-Reg. LAEA uses an LLM as a surrogate for numerical decision vectors and provides a regression-only variant, LAEA-Reg, in addition to its full regression and classification pipeline [12]. Our Direct-LLM (LAEA-style) baseline retains the core regression mechanism but replaces a fixed-dimensional vector with the source of a target program. We use this name rather than LAEA because we do not transfer LAEA’s numerical reproduction operator, classification model, or unevaluated population to the free-form program space. Historical examples and prediction. At each promotion event, the baseline sorts completed real outcomes by official utility and retains the best τ=min(50,|ℋ|)τ= (50,|H|), matching the default history limit in the released LAEA implementation. Real-valid utilities in this set are min–max normalized to [0,1][0,1], while invalid outcomes receive the common worst label of zero. For each unevaluated candidate x, the LLM receives the scientific task and objective, the retained program sources and normalized real outcomes, and the source of x. One inference is made per candidate at temperature zero. The response must contain only a scalar y^LLM(x)∈[0,1] y_LLM(x)∈[0,1] under the JSON key predicted_quality, either as a bare object or inside one JSON code fence, with no additional fields or prose. Values outside this interval are malformed. A malformed response is retried twice; if all three attempts fail, the candidate receives the worst prediction of zero. Selection and real validation. Once the history contains at least two distinct real outcome labels, the baseline promotes xpromote=argmaxx∈y^LLM(x).x_promote= _x y_LLM(x). (7) Thus, one valid and one invalid real outcome are sufficient to start regression, whereas repeated invalid outcomes are not. Before this condition holds, candidate generation follows the shared cold-start path and sends the generated candidate directly to the real evaluator. The baseline does not use a separate validity predictor, uncertainty, novelty, adaptive gates, evaluator portfolios, regional credit, or periodic unranked evaluation. Only a real-valid program may enter the algorithm population, and every completed real outcome is added to the subsequent prediction history. Appendix C Zero-Shot Reuse under Objective Reweighting We test whether a component evaluator can be reused when objective weights change, without collecting target labels. The FDTD Demux objective contains four transmission terms: T1AT_1A and T2BT_2B measure the desired routing of wavelength 1 to port A and wavelength 2 to port B, while T1BT_1B and T2AT_2A measure crosstalk into the incorrect ports. The original objective weights the two desired terms equally at (0.5,0.5)(0.5,0.5); we change these weights to (0.8,0.2)(0.8,0.2) while keeping both crosstalk penalties at −0.25-0.25. We reuse the frozen, source-calibrated component evaluator and apply the new weights directly to its predictions of these four terms, without target labels or refitting. On 36 unseen candidates, the reweighted evaluator achieves an NDCG@3 of 0.9590.959 and a normalized best regret@3 of 0.0000.000. Thus, its component representation transfers immediately to the reweighted objective. Appendix D LLM Backbone Sensitivity Table IV evaluates backbone sensitivity on PyBaMM at B=70B=70. Gemini 3.1 Pro Preview yields higher AUBCΔ and terminal improvements for both search methods. With Gemini 2.5 Flash, Janus has lower AUBCΔ than Alg.-Only but achieves a higher terminal improvement. This pattern suggests that the weaker backbone produces less useful evaluator mutations and target-program proposals early in the search, delaying improvements in proxy-guided budget allocation. As the real-outcome archive grows, the LLM receives richer task feedback and more labels become available for evaluator fitting and selection, allowing Janus to recover by the end of the search. LLM Backbone Janus Alg.-Only Advantage AUBCΔ (%) ↑ BestΔ@B (%) ↑ AUBCΔ (%) ↑ BestΔ@B (%) ↑ AUBCΔ (p) ↑ BestΔ@B (p) ↑ Gemini 3.1 Pro Preview 312.9 362.6 274.2 288.3 +38.7+38.7 +74.2+74.2 Gemini 2.5 Flash 106.7 205.5 119.6 188.4 −12.9-12.9 +17.1+17.1 TABLE IV: LLM-backbone sensitivity on PyBaMM at B=70B=70. Appendix E Token-Cost Accounting We record the prompt and completion tokens used by each method and report the average token count per completed real-evaluator call in Table V. Case Janus Alg.-Only Direct-LLM Embed-GP Direct-LLM / Janus PyBaMM 20.120.1 7.37.3 119.2119.2 17.617.6 5.9×5.9× AQM 17.117.1 5.15.1 190.1190.1 17.717.7 11.1×11.1× FDTD Demux 25.725.7 5.75.7 265.6265.6 21.621.6 10.3×10.3× Reactor 4.94.9 2.72.7 32.132.1 14.414.4 6.6×6.6× Perishable IRP 7.97.9 2.72.7 53.553.5 7.77.7 6.8×6.8× TABLE V: Average LLM tokens per completed real-evaluator call (10310^3 tokens; prompt plus completion; lower is better). Janus uses 4.94.9–25.725.7 thousand tokens per real-evaluator call across the five cases. Direct-LLM consistently uses the most tokens, requiring 5.95.9–11.1×11.1× as many tokens as Janus, with a mean ratio of 8.1×8.1×. Janus and Embed-GP have similar token counts in four cases: Janus is never more than 1.2×1.2× higher, while on Reactor it uses 2.9×2.9× fewer tokens. Alg.-Only has the lowest token count in every case; Janus uses 1.81.8–4.5×4.5× as many tokens as Alg.-Only. Appendix F Evaluator Evolution Case Studies The accuracy comparison in the main paper shows that evolved evaluators rank held-out candidates more accurately than their seed versions. The three cases below summarize the domain-relevant information introduced through evolution. PyBaMM: degradation and safety features. The evolved evaluator augments aggregate charging outcomes with separate indicators of peak temperature, lithium plating, SEI growth, voltage headroom, and time spent at high state of charge. These signals help distinguish protocols that appear similar in overall charging performance but differ in degradation risk or proximity to safety constraints. AQM: response and burst features. The evolved evaluator combines controller-structure information with controlled response probes and burst-specific measurements. It captures how marking probability and thresholds react to queue and utilization changes, while retaining goodput, tail delay, and loss as separate signals under bursty traffic. This gives the evaluator a more direct view of transient controller behavior than a single aggregate rollout score. FDTD: morphology features. The evolved evaluator supplements simulation scores with spatial descriptions of the dielectric layout, including material density, variation, mirror asymmetry, and top–bottom material imbalance. These morphology features help differentiate layouts whose coarse simulations are similar but whose spatial structure supports different routing behavior at the two output ports. Across the three domains, evaluator evolution therefore enriches the proxy with features tied to the real evaluator’s decision criteria: physical safety and degradation for charging, transient response for queue management, and spatial morphology for photonic routing.