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Physics-Informed Neural Networks for Predicting Nitrous Oxide Flux
Freddy Yu, Jashanjeet Kaur Dhaliwal, Subhadeep Chakraborty
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Abstract
Abstract:Nitrous oxide (N$_2$O) is the dominant ozone-depleting substance emitted in the 21st century, and the third largest contributor to anthropogenic greenhouse gases due to its high potency and long atmospheric lifetime, with more than 70% of N$_2$O emissions occurring as a result of agricultural processes. Current approaches to predicting N$_2$O flux emissions include process-based models such as DayCent and Cycles, as well as classical AI models, but the application of Physics-Informed Neural Networks (PINNs) to predicting N$_2$O flux emissions is largely underexplored. Our paper draws upon the mechanistic equations that underlie the DayCent family of process-based models to construct a rigorously derived, literature-traceable physics residual. We then build and train an MLP-based PINN on a multi-site agricultural dataset spanning four geographically distinct US agricultural sites. Across all tested values of the physics loss weighting hyperparameter $\lambda$, our PINN consistently and substantially outperformed uncalibrated Cycles simulation (R$^2=0.01$), with our MLP baseline achieving mean R$^2=0.411$ across ten random seeds. Physics constraints consistently degrade model performance in holdout validation, with marginal degradation at low $\lambda$ and significant degradation at high $\lambda$, but consistently improve model performance and reduce performance variability in leave-one-site-out validation. This suggests that physics constraints sacrifice in-distribution accuracy for out-of-distribution robustness, anchoring the model toward biogeochemically plausible behavior on unfamiliar soil conditions --- though cross-site generalization remains challenging, with negative R$^2$ across all seeds and $\lambda$ values on our geographically distinct held-out site.
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Physics-Informed Neural Networks for Predicting Nitrous Oxide Flux Freddy Yu Harvard University freddyyu@college.harvard.edu Jashanjeet Kaur Dhaliwal Department of Biosystems Engineering and Soil Science University of Tennessee jdhaliwa@utk.edu Subhadeep Chakraborty Department of Mechanical and Aerospace Engineering University of Tennessee schakrab@utk.edu Alternate email: freddyyu99@gmail.com Abstract Nitrous oxide (N2O) is the dominant ozone-depleting substance emitted in the 21st century, and the third largest contributor to anthropogenic greenhouse gases due to its high potency and long atmospheric lifetime, with more than 70% of N2O emissions occurring as a result of agricultural processes. Current approaches to predicting N2O flux emissions include process-based models such as DayCent and Cycles, as well as classical AI models, but the application of Physics-Informed Neural Networks (PINNs) to predicting N2O flux emissions is largely underexplored. Our paper draws upon the mechanistic equations that underlie the DayCent family of process-based models to construct a rigorously derived, literature-traceable physics residual. We then build and train an MLP-based PINN on a multi-site agricultural dataset spanning four geographically distinct US agricultural sites. Across all tested values of the physics loss weighting hyperparameter λ, our PINN consistently and substantially outperformed uncalibrated Cycles simulation (R=20.01^2=0.01), with our MLP baseline achieving mean R=20.411^2=0.411 across ten random seeds. Physics constraints consistently degrade model performance in holdout validation, with marginal degradation at low λ and significant degradation at high λ, but consistently improve model performance and reduce performance variability in leave-one-site-out validation. This suggests that physics constraints sacrifice in-distribution accuracy for out-of-distribution robustness, anchoring the model toward biogeochemically plausible behavior on unfamiliar soil conditions — though cross-site generalization remains challenging, with negative R2 across all seeds and λ values on our geographically distinct held-out site. 1 Introduction Among all greenhouse gases, nitrous oxide (N2O) is the third-most important after carbon dioxide (CO2) and methane (CH4) (Tian et al., 2024), and the dominant ozone-depleting substance emitted in the 21st century (Ravishankara et al., 2009). Nitrous oxide emissions have grown by an unprecedented 40% since 1980 (Tian et al., 2024), with a global warming potential (GWP) at about 273 times that of carbon dioxide (Forster et al., 2021), and a long atmospheric lifetime of approximately 117 years (Prather and Wilson, 2026). Recent growth in N2O emissions has exceeded some of the highest projected climate scenarios (Tian et al., 2020), highlighting the need for improved predictive capability. More than 70% of nitrous oxide emissions occur as a result of agricultural processes (Tian et al., 2024), making accurate quantification of agricultural N2O flux particularly essential for greenhouse gas accounting and climate mitigation efforts. Strengthening our ability to quantify nitrous oxide emissions could yield significant downstream benefits, such as more informed policy-making, better-targeted agricultural interventions to reduce emissions, and more accurate climate projections. However, direct measurement of N2O flux emissions is very challenging (Pattey et al., 2007), in part because N2O atmospheric concentrations are low enough to lie outside the detection range of many analytical techniques (Rapson and Dacres, 2014). Static chambers remain the most common method for measuring nitrous oxide fluxes from agricultural soils (de Klein et al., 2020; Rapson and Dacres, 2014), but chamber measurements are limited in their spatial representativeness (Bastos et al., 2021; Gu et al., 2013; Rochette et al., 2010), since individual chambers typically measure emissions over small plot-scale areas (Wang et al., 2001) and may require extensive deployment to characterize heterogeneous agricultural landscapes (Charteris et al., 2020). Moreover, soil nitrous oxide emissions are inherently episodic (Yamulki et al., 2000), highly variable across both time (Phillips et al., 2013) and space (Parkin, 1987), and subject to sudden, unpredictable impacts from environmental factors like rain (Jones et al., 2011) and nitrogen fertilization (Phillips et al., 2013), making chamber estimations particularly challenging (Charteris et al., 2020) and more prone to biases due to hotspots (Brown et al., 2024). Using chambers to measure N2O flux also tends to be either fairly costly, as in the case of automatic chambers (Mastepanov, 2026), or very labor-intensive, as in the case of manual chambers (Hensen et al., 2013). Two viable and important alternatives to directly measuring N2O flux emissions are to use either AI models or process-based models (like DayCent and Cycles) to simulate or predict these emissions, based on input variables that can be more easily measured. However, both options have their own shortcomings as well. As discussed further in Section 3, process-based models generally struggle to accurately capture the complex soil biogeochemical processes that govern N2O production across chemically and climatically diverse soil systems without extensive site-specific calibration (Butterbach-Bahl et al., 2013; Sharma et al., 2026; Joshi et al., 2024). Process-based models also often require extensive calibration on site-specific information and experimental observations (Gabbrielli et al., 2024), limiting their portability across locations. DayCent has been shown to struggle with estimating nitrous oxide emissions in particular (Del Grosso et al., 2008), with poor agreement between DayCent-predicted and observed soil N2O emissions (McClelland et al., 2021). This is almost certainly in part due to nitrous oxide’s tendency to emit in spontaneous, sporadic "hot spots" and "hot moments" (Kravchenko et al., 2017; Turner et al., 2016; Wagner-Riddle et al., 2020), given that process-based models (including DayCent) are reported to systematically under-predict emissions when actual emissions are high, i.e., during hotspots (Gaillard et al., 2018), and to struggle with the temporal dynamics of N2O emissions (Jarecki et al., 2008). The application of machine learning to N2O prediction remains relatively nascent (Sharma et al., 2026), with comparatively fewer studies systematically characterizing its limitations relative to the process-based modeling literature. Classical machine learning models like Random Forest have demonstrated early promise for N2O prediction (Philibert et al., 2013). However, initial findings suggest that N2O predictions made by ML models may suffer from many of the same issues as those made by process-based models. Like process-based models, traditional machine learning models similarly struggle to capture the complex biogeochemical processes that drive nitrous oxide emissions (Gnisia et al., 2025), and struggle to generalize beyond their training domain (Sharma et al., 2026). In this paper, we propose and implement a Physics-Informed Neural Network (PINN) that combines data-driven prediction with biogeochemical constraints derived from the DayCent family of process-based models. First suggested by Raissi et al. (2019), PINNs are neural networks whose loss functions contain two components: the data loss function (typically the MSE of model predictions relative to measured data) and the physics loss function (the MSE of model predictions relative to values predicted by biogeochemical equations). In the case of N2O predictions, these equations represent the real-world biogeochemical conditions that constrain nitrous oxide emissions. Vemuri (2024) previously applied this framework to N2O prediction — the only such attempt known to the authors — finding that physics-informed loss terms may improve predictive accuracy. However, their physics residual relied on generic Michaelis-Menten enzyme kinetics rather than equations derived specifically for soil nitrogen biogeochemistry, their dataset comprised 2,246 observations from two geographically similar Midwestern US sites (Saha et al., 2021), and their evaluation included no leave-one-site-out validation or comparison against process-based model baselines. In this paper, we address these limitations directly: we derive our physics residual from the mechanistic equations underlying the DayCent family of process-based models, train and evaluate on 8,271 daily observations across four geographically distinct US sites, and benchmark against uncalibrated Cycles simulation as a process-based baseline. We demonstrate that our PINN substantially outperforms uncalibrated Cycles simulation in holdout validation, and find that while the physics residual hurts in-distribution performance, it consistently improves performance and reduces variability in leave-one-site-out validation — suggesting a tradeoff between in-distribution accuracy and out-of-distribution robustness. 2 Methodology 2.1 PINN Background The loss function of PINNs is given by: ℒ=ℒdata+λℒphysicsL=L_data+ _physics (1) Variable Description Units Source ℒL total loss g2 N ha-2 d-2 ℒdataL_data data loss g2 N ha-2 d-2 Equation 2 λ weight of the physics loss residual unitless hyperparameter (relative to data loss) ℒphysicsL_physics physics loss residual g2 N ha-2 d-2 Equation 3 Our model’s data loss is calculated as the mean-squared error (MSE) of the model-predicted N2O values, relative to the measured ("actual") N2O values, over n training samples, following Raissi et al. (2019). ℒdata=1n∑i=1n(ϕN2Oi−ϕ^N2Oi)2L_data= 1n _i=1^n ( _N_2O_i- φ_N_2O_i )^2 (2) Variable Description Units Source ℒdataL_data data loss g2 N ha-2 d-2 ϕN2O _N_2O measured N2O flux g N ha-1 d-1 DSFAS data ϕ^N2O φ_N_2O model-predicted N2O flux g N ha-1 d-1 model Similarly, our model’s physics loss is calculated as the mean-squared error (MSE) of the model-predicted N2O values, relative to the values predicted by our DayCent-derived equations, following Raissi et al. (2019). ℒphysics=1n∑i=1n(ϕ~N2Oi−ϕ^N2Oi)2L_physics= 1n _i=1^n ( φ_N_2O_i- φ_N_2O_i )^2 (3) Variable Description Units Source ℒphysicsL_physics physics loss residual g2 N ha-2 d-2 ϕ~N2O φ_N_2O physics-predicted N2O flux g N ha-1 d-1 Equation 9 ϕ^N2O φ_N_2O model-predicted N2O flux g N ha-1 d-1 model prediction 2.2 Physics Loss Residual We now lay out the foundational equations that we use to calculate the physics-predicted N2O values, drawing upon three primary papers, which form the fundamental basis of the Nitrogen Trace Gas Submodel of DayCent as described in Part 2, Section 3.5 of Hartman et al. (2018): Parton et al. (1996), Del Grosso et al. (2000), and Parton et al. (2001). We additionally draw upon Hartman and Parton (2019) for the equations of specific graphs when they are not provided in the three aforementioned papers. Note that DayCent is a constantly evolving process model, and that the equations our paper uses may no longer be in use in newer generations of DayCent. As the Preface of Hartman et al. (2018) states directly: "The model attempts to document multiple versions of the DayCent model that differ in the set processes they include, the way these processes are calculated, and the format of input and output files. Different versions emerge as separate research groups modify the model to meet the requirements of their research and applications." Following Parton et al. (2001), the equations for nitrous oxide flux as a result of nitrification are given by: ϕNO3=Netmin⋅K1+Kmax⋅NH4⋅Fn(Ts)⋅Fn(WFPS)⋅Fn(pH) _NO_3=Net_min· K_1+K_max· NH_4· F_n(T_s)· F_n(WFPS)· F_n(pH) (4) Variable Description Units Source ϕNO3 _NO_3 soil nitrification flux g N m-2 d-1 NetminNet_min daily net N mineralization g N m-2 d-1 Cycles K1K_1 fraction of NetminNet_min assumed to be nitrified each day unitless constant at 0.20 NH4NH_4 model-derived soil ammonium concentration g N m-2 Cycles KmaxK_max maximum fraction of NH4+NH_4^+ nitrified d-1 constant at 0.10 Fn(Ts)F_n(T_s) effect of soil temperature on nitrification unitless Equation 10 Fn(WFPS)F_n(WFPS) effect of water-filled pore space on nitrification unitless Equation 12 Fn(pH)F_n(pH) effect of soil pH on nitrification unitless Equation 17 ϕN2O,nit=K2⋅ϕNO3 _N_2O,nit=K_2· _NO_3 (5) Variable Description Units Source ϕN2O,nit _N_2O,nit N2O flux from nitrification g N m-2 d-1 K2K_2 fraction of nitrified N lost as ϕN2O,nit _N_2O,nit unitless constant at 0.02 ϕNO3 _NO_3 soil nitrification flux g N m-2 d-1 Equation 4 Following Parton et al. (1996), the equation for total nitrogen gas fluxes (including both nitrogen and nitrous oxide) as a result of denitrification is given below. (Note that this equation also appears in Del Grosso et al. (2000), but with no modifications, so we cite Parton et al. (1996) as the original source.) ϕN,den=min(ϕmax(NO3),ϕmax(CO2))⋅Fd(WFPS) _N,den= min( _max(NO_3), _max(CO_2))· F_d(WFPS) (6) Variable Description Units Source ϕN,den _N,den total N gas fluxes from denitrification g N ha-1 d-1 ϕmax(NO3) _max(NO_3) max ϕN,den _N,den for a given NO3 level, assuming high CO2 g N ha-1 d-1 Equation 18 ϕmax(CO2) _max(CO_2) max ϕN,den _N,den for a given respiration rate, assuming high NO3 g N ha-1 d-1 Equation 19 Fd(WFPS)F_d(WFPS) effect of WFPS on denitrification unitless Equation 21 Following Del Grosso et al. (2000), the following two equations for the ratio of nitrogen gas to nitrous oxide gas are given by: RN2/N2O=Fr(NO3/CO2)⋅Fr(WFPS)R_N_2/N_2O=F_r(NO_3/CO_2)· F_r(WFPS) (7) Variable Description Units Source RN2/N2OR_N_2/N_2O the ratio of N2 to N2O gases produced from denitrification unitless Fr(NO3/CO2)F_r(NO_3/CO_2) NO3-to-respiration ratio factor unitless Equation 22 Fr(WFPS)F_r(WFPS) effect of water-filled pore space on RN2/N2OR_N_2/N_2O unitless Equation 39 Using this equation, we can then calculate the amount of nitrous oxide gas produced from denitrification. ϕN2O,den=ϕN,den1+RN2/N2O _N_2O,den= _N,den1+R_N_2/N_2O (8) Variable Description Units Source ϕN2O,den _N_2O,den N2O flux from denitrification g N ha-1 d-1 ϕN,den _N,den total N gas fluxes from denitrification g N ha-1 d-1 Equation 6 RN2/N2OR_N_2/N_2O the ratio of N2 to N2O gases produced from denitrification unitless Equation 7 Then, finally, by drawing upon all of the prior equations: ϕ~N2Oi=αm2→ha⋅ϕN2O,niti+ϕN2O,deni φ_N_2O_i= _m^2→ ha· _N_2O,nit_i+ _N_2O,den_i (9) Variable Description Units Source ϕ~N2O φ_N_2O physics-predicted N2O flux g N ha-1 d-1 αm2→ha _m^2→ ha units conversion factor from m-2 to ha-1 m2 ha-1 constant at 10410^4 ϕN2O,nit _N_2O,nit N2O flux from nitrification g N m-2 d-1 Equation 5 ϕN2O,den _N_2O,den N2O flux from denitrification g N ha-1 d-1 Equation 8 The other equations we used, as well as their origins in the research literature, can be found in Section A.1. 2.3 Dataset All of the variables that we use in our work can be categorized into one of three categories: 1. "Input Variables": The "starting" variables whose data is either directly measured or directly simulated by Cycles. These are also the same predictor variables that are fed directly into the model as input features. 2. "Intermediate Variables": The variables that are derived from the input variables using the equations described in Section 2.2 and Section A.1 3. "Final Variables": The variables that are directly used to calculate the model loss and evaluate its performance; namely: ϕN2O _N_2O (measured N2O flux), ϕ^N2O φ_N_2O (model-predicted N2O flux), and ϕ~N2O φ_N_2O (physics-predicted N2O flux). The following table is an exhaustive list of the 10 input variables we used. Table 1: Input Variables Variable Description Units Source NO3NO_3 soil nitrate concentration mg N kg-1 Cycles NH4NH_4 model-derived soil ammonium concentration g N m-2 Cycles pHpH soil pH in the 2nd soil layer unitless static site measurement TsT_s average soil temperature of soil layers 2 and 3 ∘C Cycles Ta,maxT_a,max maximum monthly air temperature ∘C Cycles-approximated %S\%S percentage of soil that is sand unitless site survey %C\%C percentage of soil that is clay unitless site survey NetminNet_min daily net N mineralization g N m-2 d-1 Cycles VWClVWC_l volumetric water content of soil layer l m3 m-3 Cycles BDlBD_l soil bulk density of soil layer l g cm-3 Cycles ϕN2O _N_2O, our measured N2O flux, is taken from Robertson and Saha (2026), an online living dataset hosted by the KBS LTER (Kellogg Biological Station Long-Term Ecological Research) program at Michigan State University, and part of the USDA-sponsored Data Science for Food and Agricultural Systems (DSFAS) project. We accessed the data on June 19, 2026; the dataset has likely been updated and modified since then. Our dataset consists of data drawn from four sites (some of which were treated with various treatment types), enumerated as follows: 1. An agricultural field near Ames, IA 2. KBS GLBRC, Hickory Corners, MI, USA (Treatments G1, G2, G3, G4) 3. Kelly North, Iowa State Research Farm, Ames, Iowa 4. Cook Agronomy Farm, No-till (AmeriFlux US-RC1) (Treatments NT and CT) With the assistance of Robertson and Saha project collaborators, we traced through tillage history to characterize Sites 1 and 3 as conventionally tilled, and Site 2 as untilled. Site 4 with treatment "CT" was conventionally tilled, and Site 4 with treatment "NT" was not tilled. For Equation 39 as in Del Grosso et al. (2000), we equate "conventionally tilled" with "repacked", and "not tilled" as "intact". In cases where our equations expect site-specific values, but our data only provides layer-specific values, we take the weighted average across layers 2 and 3 (under Hartman and Parton (2019)’s layer convention), where the weight for each layer is the layer thickness. This is the same approach we use in, for instance, Equation 13a. Our measured N2O flux data was preprocessed as follows: Sub-daily measurements were aggregated to daily values to match the daily time step of the Cycles model. Flux measurements were then averaged across field replications, as Cycles does not simulate replication. For sites with measurements at multiple landscape elevations, only data from the highest elevation were retained to maintain consistency with the one-dimensional structure of the Cycles model. This preprocessing was performed prior to delivery of the dataset used in this work. Our data on the sand and clay percentages (%S and %C, respectively) was sourced from gSSURGO/SoilGrids or original site publications. In our variable tables, we hereafter simply list the source as “site survey". With the sole exception of soil pH (which was a static site-level measurement), all of our input variables are directly simulated by Cycles. Although measured data on most of the input variables we need was available via the DSFAS project dataset, the measured data had serious gaps. Some sites lacked measurements of certain key variables like net mineralization and soil bulk density, and even when present, the measurements were often taken at varying depths. Rather than taking a "mix-and-match" approach of combining measured data taken from the DSFAS dataset with simulated data taken from Cycles, we instead opted to go with the more cohesive, consistent approach of using entirely Cycles-simulated data. Our reasoning is as follows: Our DayCent-derived equation chain maps a joint soil state (NO3, NH4, TS, VWC, BD, %S, %C…) to a single predicted N2O value. This residual is only physically meaningful if the input vector at each timestep is a jointly realizable state, i.e., something that could actually co-occur in one soil at one time. Our approach of using entirely Cycles-simulated data guarantees this condition, because it generated every driver under one internally-coupled dynamics. For instance, Cycles-simulated NO3 and VWC data at day t are produced by the exact same simulated moisture/temperature trajectory. Gap-filling (the alternative mix-and-match approach) would break this condition: By filling a single variable at time t with both Cycles-simulated data and measured data, we’d be feeding the equations a state vector that no coherent soil ever occupied. Moreover, filling gaps in measured data with Cycles-simulated data would render feature provenance (i.e., data sourcing) as a function of site. More specifically: Measured data from the DSFAS dataset is available for some sites but not for other sites, so if our model were to struggle in leave-one-site-out validation, it’d be impossible to distinguish between the model’s struggle to generalize geographically and the model’s struggle to generalize between simulated and measured data. For instance, suppose our model is trained on Site A, which has mostly measured data, and then tested on Site B, which has mostly Cycles-simulated data. The model’s struggle to generalize across geographic sites would appear externally identical to its struggle to generalize between the measured data of Site A and the simulated data of Site B. In summary: augmenting the DSFAS measured data with simulated data would confound provenance with geographic site identity. 2.4 Model Architecture & Training We use PyTorch to implement a three-layer MLP (10→64→32→110→ 64→ 32→ 1), with ReLU activation and dropout (p=0.3) applied after each of the first two linear layers. Our model utilizes the Adaptive Moment Estimation (Adam) optimizer with learning rate 0.001. As mentioned in Section 2.1, we use MSE for all of our loss functions. Our model also uses early stopping with patience=20 to mitigate overfitting on training data. Our findings as discussed in Section 3 were calculated as the mean across ten random seeds (41 through 50) unless explicitly stated otherwise. We make use of two distinct model evaluation techniques: holdout validation, and leave-one-site-out validation. In the case of holdout validation, we use a random 80/20 training/testing data split, and then further divide the training data using an 80/20 training/validation data split. The training data is the data that the model is actually trained on, while the validation dataset is used to evaluate model performance for the early stopping mechanism. This is necessary because using the testing dataset directly for early stopping would be a form of data leakage. In the case of leave-one-site-out validation, we use data from Site 4 as our testing data, since Site 4 is most distinct from the rest of our dataset, both geographically and in terms of soil characteristics (i.e., pH, soil texture, etc., as discussed further in Section 3), and thus, the most demanding test of geographic generalizability. Sites 1–3 are split 80/20 into training and validation sets, with the validation set used exclusively for early stopping. 3 Results & Discussion 3.1 Model Performance Relative to Cycles Unless otherwise noted, per-site results in this subsection are reported for seed 41 and λ=0λ=0; aggregate results across all seeds are provided in Section 3.2 and Appendix A.3. Our baseline model (λ=0λ=0) outperformed Cycles dramatically for every site, achieving overall R=20.423^2=0.423 on seed 41 against Cycles’ mean of R=20.01^2=0.01, with even the worst-performing configuration (λ=1λ=1, mean R=20.090^2=0.090) outperforming Cycles by nearly an order of magnitude. We also find, perhaps unsurprisingly, that our model performed best on Site 2, for which it had the most training data by a wide margin. By comparison, its test R2 value for Sites 1 and 4 were worse by more than 0.15. Table 2: Per-site R2 summary (holdout validation, seed=41, λ=0) Site Dataset Size Split Model R2 Cycles R2 1 435 Train 0.228 -0.244 Test 0.264 -0.407 2 6,468 Train 0.490 0.122 Test 0.490 0.181 3 224 Train 0.645 -0.376 Test -0.299 -0.829 4 1,144 Train 0.329 -0.210 Test 0.333 -0.259 Overall 8,271 Train 0.420 0.011 Test 0.442 0.003 All 0.423 0.010 Our model scores a remarkably low R=2−0.299^2=-0.299 on Site 3, but this can be explained by the occurrence of a single outlier. Near late July of 2006 in our data, our model predicted a spike of nearly 100 g N ha-1 d-1, more than three times the measured flux of approximately 30 g N ha-1 d-1. Because R2 penalizes errors in proportion to their squared magnitude, a single sufficiently large prediction error can dominate the metric. When this outlier is removed, Site 3 test R2 jumps from −0.299-0.299 to 0.5180.518, consistent with performance at our other sites. For further analysis, we now turn our attention to the two sites that both our model and Cycles performed best and worst at, respectively: Site 2 and Site 1. Figure 1: Site 2 time series, April–October 2014 (most active 6-month emission period) Figure 1 illustrates N2O dynamics at Site 2 during the most active emission period (April–October 2014). Both our model and Cycles struggled with N2O’s characteristic pattern of near-zero background emissions punctuated by sudden, volatile hotspots. Cycles consistently underestimated peak emissions by almost a full order of magnitude. Our model generally matched the timing of emission events more closely, though it still somewhat underestimated peak magnitudes. At Site 1, both our model and Cycles failed to capture most N2O hotspots (see Figure 4 in the Appendix). Our model overestimated background emissions relative to Cycles, but neither approach responded adequately to peak emission events. We speculate this reflects underrepresentation in training data, as Site 1 comprised only 4358271=5.26% 4358271=5.26\% of the total dataset. (See Figures 5–7 in the Appendix for the complete time series.) 3.2 Physics Residual: In-Distribution vs. Out-of-Distribution Performance Figure 2: R2 across λ sweep (holdout validation). Individual seed trajectories (seeds 41–50). Physics constraints consistently degrade in-distribution performance, with degradation increasing at higher λ. Seed 43 exhibits notably high sensitivity at large λ values. Figure 3: R2 across λ sweep (leave-one-site-out validation). Mean ± 1 std across seeds 41–50. Physics constraints consistently improve out-of-distribution performance, peaking at λ=0.7λ=0.7. Values below −80-80 are clipped; see Table 6 in the Appendix for complete values. From Figure 3, we see that in the case of holdout validation, the physics constraints imposed by our physics loss residual ℒphysicsL_physics consistently hurt model performance across all tested values of λ, with model performance degradation increasing as physics weight λ increases. We additionally see that, as λ increases, the mean standard deviation across the ten seeds increases, suggesting that a higher physics weight results in greater performance variability. (See Table 5 in the Appendix for complete results.) This general pattern is flipped in the case of leave-one-site-out validation. From Figure 3, we see that although our model’s performance is severely negative, with negative R2 values across all seeds and λ values, the general trend is the opposite of what we saw in holdout validation. As λ increases, mean R2 increases and mean standard deviation decreases, suggesting that a higher physics weight results in improved model performance and less performance variability. Our model’s performance peaks at λ=0.7λ=0.7, where mean R=2−12.597^2=-12.597, a substantial improvement from the MLP baseline λ=0.00λ=0.00, where mean R=2−71.672^2=-71.672. Moreover, the mean standard deviation at λ=0.7λ=0.7 reaches its lowest value of 11.28111.281, insufficient to explain the significant gap in performance between λ=0.7λ=0.7 and λ=0.0λ=0.0 means. (See Table 6 in the Appendix for complete results.) To explain this apparent contradiction, we turn to recent findings from Del Grosso and Ogle (2026) discussing Sharma et al. (2026): that process-based models accurately represent the drivers of N2O emissions (soil water content, mineral N, temperature) but systematically struggle to predict the fraction of cycled N (produced from denitrification) that actually gets emitted as N2O. This is because the nitrification and denitrification processes that govern N2O production are sufficiently complex that current equations necessarily rely on simplifying assumptions. For instance, the use of water-filled pore space (WFPS) as a factor in N2O production is generally considered non-ideal at best (Farquharson and Baldock, 2008), but functions based on water filled pore space are still widely used (such as in DayCent), in the absence of widely validated alternatives. Beyond conceptual limitations like WFPS, our physics residual follows well-established research literature in using fixed biogeochemical constants like K1=0.20,K2=0.02,Kmax=0.10K_1=0.20,K_2=0.02,K_max=0.10, etc. Ultimately, these constants are widely accepted but nonetheless flawed simplifications that hinder our physics residual’s ability to accurately model real-world physical constraints. In summary, even a perfectly implemented physics residual with perfectly derived equations would be expected to provide weak or inconsistent regularization signal on heterogeneous multi-site data, because the biogeochemical constants themselves are the source of uncertainty. This simultaneously explains the poor performance of both the uncalibrated Cycles model and our PINN’s physics residual. Consequently, we suspect that the physics residual hurts in-distribution (i.e., in holdout validation) because, as suggested by Del Grosso and Ogle (2026), fixed constants are too coarse to improve local accuracy, and the data alone is a much better predictor of N2O emissions. However, the physics residual helps out-of-distribution (i.e., in leave-one-site-out validation) because even imperfect physics constraints are better than nothing when the model encounters unfamiliar soil conditions. More precisely: the same biogeochemical constants that are too imprecise to improve model performance in holdout validation, are simultaneously precise enough to prevent the model from making biogeochemically implausible predictions on unseen soil types. In the case of our leave-one-site-out validation testing, the soil characteristics of our left-out test site, Site 4, are substantially different from those of our other 3 training sites. Site 4 had an average soil pH of 5.2 and average soil texture of 21.0% clay and 11.3% sand, where the other 3 sites had an average soil pH of 7.5 and average soil texture of 12.3% clay and 61.4% sand. In the face of such unfamiliar soil conditions, our physics residual may serve to help anchor the model to biogeochemically plausible behavior. If this reasoning is correct, then we conclude that, while the physics residuals of PINNs are not magic performance boosters, they may serve as an important fallback: when the model needs to generalize across geographically distinct sites, the physics residual offers the tradeoff of sacrificing in-distribution accuracy for out-of-distribution robustness. 4 Conclusion In summary, our model achieves mean R2 = 0.411 in holdout validation, dramatically outperforming Cycles (mean R2 = 0.010), but struggles significantly in leave-one-site-out validation, achieving negative R2 at every value of λ across all ten seeds, suggesting that geographic generalizability remains challenging. Our model’s best performance in holdout validation occurred at λ≈0λ≈ 0, while its best performance in leave-one-site-out validation occurred at λ≈0.7λ≈ 0.7 — an approximately 6× improvement in mean R2 over the MLP baseline — suggesting that physics constraints sacrifice in-distribution accuracy for out-of-distribution robustness, anchoring the model toward biogeochemically plausible behavior on unfamiliar soil conditions. Our paper does have notable methodological weaknesses. Firstly, with the sole exception of pH and measured N2O, all of our data was Cycles-simulated. Cycles-derived inputs carry whatever systematic biases or errors Cycles has, meaning that from the very start, our PINN’s predictive accuracy was bounded by how well Cycles represents the true soil state. Secondly, our paper does not make use of collocation points as Raissi et al. (2019) did in their original PINN. Our setting does not naturally lend itself to collocation points given its discrete daily timestep structure, meaning our physics loss is enforced only at observed data points, which may limit the regularization effect of the physics constraint. Thirdly, our physics residual relies on fixed biogeochemical constants (e.g., K1=0.20K_1=0.20, K2=0.02K_2=0.02, Kmax=0.10K_max=0.10) that are widely accepted simplifications but do not vary across sites or soil conditions, limiting the residual’s ability to provide meaningful regularization signal on heterogeneous multi-site data. Fourthly, although our data is sourced from Cycles, our equations are derived from the DayCent family; this mismatch may result in the physics residual’s performance being hampered by conflating the internal biases of two different process-based models. Finally, our dataset was not evenly geographically distributed — our assessment of the model’s overall performance was heavily dominated by its performance at Site 2, due to Site 2’s overrepresentation in the data (constituting 6,468 of 8,271 points, or 78.2% of our entire dataset). Future work should experiment with different model architectures beyond MLPs (such as PINNsformers as in Zhao et al. (2024) or S-Pformers as in Arni and Blanco (2025)), as well as different process-based models and physics residuals (such as implementing Cycles equations for a more direct comparison with uncalibrated Cycles performance). We furthermore believe that more extensive leave-one-site-out experimentation, particularly with larger and more geographically diverse datasets, could help validate our hypothesis that physics residuals improve out-of-distribution robustness. Acknowledgments and Disclosure of Funding F.Y. was supported by the 2026 Student Mentoring and Research Training (SMaRT) program provided by The Science Alliance, which is a Tennessee Higher Education Commission (THEC) center of excellence administered by The University of Tennessee-Oak Ridge Innovation Institute (UT-ORII). This work was supported in part by USDA NIFA, Award #2023-67021-40008. We thank the DSFAS project team at KBS LTER for providing the N2O flux dataset used in this work. References R. Arni and C. Blanco (2025) Physics-Informed Neural Networks with Fourier Features and Attention-Driven Decoding. arXiv. Note: arXiv:2510.05385 [cs.LG] External Links: Link, Document Cited by: §4. L. M. Bastos, C. W. Rice, P. J. Tomlinson, and D. Mengel (2021) Untangling soil-weather drivers of daily N2O emissions and fertilizer management mitigation strategies in no-till corn. Soil Science Society of America Journal 85 (5), p. 1437–1447 (en). Note: _eprint: https://acsess.onlinelibrary.wiley.com/doi/pdf/10.1002/saj2.20292 External Links: ISSN 1435-0661, Link, Document Cited by: §1. S. E. Brown, C. Wagner-Riddle, and B. Conrad (2024) Low-power flux gradient measurements for quantifying the impact of agricultural management on nitrous oxide emissions. Agricultural and Forest Meteorology 353, p. 110027. External Links: ISSN 0168-1923, Link, Document Cited by: §1. K. Butterbach-Bahl, E. M. Baggs, M. Dannenmann, R. Kiese, and S. Zechmeister-Boltenstern (2013) Nitrous oxide emissions from soils: how well do we understand the processes and their controls?. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences 368 (1621), p. 20130122 (eng). External Links: ISSN 1471-2970, Document Cited by: §1. A. F. Charteris, D. R. Chadwick, R. E. Thorman, A. Vallejo, C. A.M. de Klein, P. Rochette, and L. M. Cárdenas (2020) Global Research Alliance N2O chamber methodology guidelines: Recommendations for deployment and accounting for sources of variability. Journal of Environmental Quality 49 (5), p. 1092–1109 (en). Note: _eprint: https://acsess.onlinelibrary.wiley.com/doi/pdf/10.1002/jeq2.20126 External Links: ISSN 1537-2537, Link, Document Cited by: §1. E. A. Davidson and S. E. Trumbore (1995) Gas diffusivity and production of CO2 in deep soils of the eastern Amazon. Tellus B 47 (5), p. 550–565 (en). External Links: ISSN 0280-6509, 1600-0889, Link, Document Cited by: §A.1, §A.1, §A.1, §A.1. C. A. M. de Klein, M. A. Alfaro, D. Giltrap, C. F. E. Topp, P. L. Simon, A. D. L. Noble, and T. J. van der Weerden (2020) Global Research Alliance N2O chamber methodology guidelines: Statistical considerations, emission factor calculation, and data reporting. Journal of Environmental Quality 49 (5), p. 1156–1167 (en). Note: _eprint: https://acsess.onlinelibrary.wiley.com/doi/pdf/10.1002/jeq2.20127 External Links: ISSN 1537-2537, Link, Document Cited by: §1. S. J. Del Grosso, A. D. Halvorson, and W. J. Parton (2008) Testing DAYCENT Model Simulations of Corn Yields and Nitrous Oxide Emissions in Irrigated Tillage Systems in Colorado. Journal of Environmental Quality 37 (4), p. 1383–1389 (en). Note: _eprint: https://acsess.onlinelibrary.wiley.com/doi/pdf/10.2134/jeq2007.0292 External Links: ISSN 1537-2537, Link, Document Cited by: §1. S. J. Del Grosso, W. J. Parton, A. R. Mosier, D. S. Ojima, A. E. Kulmala, and S. Phongpan (2000) General model for N2O and N2 gas emissions from soils due to dentrification. Global Biogeochemical Cycles 14 (4), p. 1045–1060 (en). Note: _eprint: https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/1999GB001225 External Links: ISSN 1944-9224, Link, Document Cited by: §A.1, §A.1, §A.1, §A.1, §A.1, §2.2, §2.2, §2.2, §2.3. S. Del Grosso and S. Ogle (2026) Combining process-based models with machine learning ensemble improves soil nitrous oxide estimates. Proceedings of the National Academy of Sciences 123 (16), p. e2605414123. External Links: Link, Document Cited by: §3.2, §3.2. R. Farquharson and J. Baldock (2008) Concepts in modelling N2O emissions from land use. Plant and Soil 309 (1-2), p. 147–167 (en). External Links: ISSN 0032-079X, 1573-5036, Link, Document Cited by: §3.2. P. Forster, T. Storelvmo, K. Armour, W. Collins, J.-L. Dufresne, D. Frame, D.J. Lunt, T. Mauritsen, M.D. Palmer, M. Watanabe, M. Wild, and H. Zhang (2021) Chapter 7: The Earth’s Energy Budget, Climate Feedbacks, and Climate Sensitivity. In In Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change, V. Masson-Delmotte, P. Zhai, A. Pirani, S.L. Connors, C. Péan, S. Berger, N. Caud, Y. Chen, L. Goldfarb, M.I. Gomis, M. Huang, K. Leitzell, E. Lonnoy, J.B.R. Matthews, T.K. Maycock, T. Waterfield, O. Yelekçi, R. Yu, and B. Zhou (Eds.), p. 923–1054 (en). External Links: Link, Document Cited by: §1. M. Gabbrielli, M. Allegrezza, G. Ragaglini, A. Manco, L. Vitale, and A. Perego (2024) A Review of the Main Process-Based Approaches for Modeling N2O Emissions from Agricultural Soils. Horticulturae 10 (1), p. 98 (en). External Links: ISSN 2311-7524, Link, Document Cited by: §1. R. K. Gaillard, C. D. Jones, P. Ingraham, S. Collier, R. C. Izaurralde, W. Jokela, W. Osterholz, W. Salas, P. Vadas, and M. D. Ruark (2018) Underestimation of N2O emissions in a comparison of the DayCent, DNDC, and EPIC models. Ecological Applications 28 (3), p. 694–708 (en). Note: _eprint: https://esajournals.onlinelibrary.wiley.com/doi/pdf/10.1002/eap.1674 External Links: ISSN 1939-5582, Link, Document Cited by: §1. G. Gnisia, J. Weik, R. Ruser, L. Essich, I. Lewandowski, and A. Stein (2025) Machine learning-based prediction of nitrous oxide emissions from arable farming: Exploring management practices as predictor variables. Ecological Indicators 172, p. 113233. External Links: ISSN 1470-160X, Link, Document Cited by: §1. J. Gu, B. Nicoullaud, P. Rochette, A. Grossel, C. Hénault, P. Cellier, and G. Richard (2013) A regional experiment suggests that soil texture is a major control of N2O emissions from tile-drained winter wheat fields during the fertilization period. Soil Biology and Biochemistry 60, p. 134–141. External Links: ISSN 0038-0717, Link, Document Cited by: §1. M. D. Hartman and W. J. Parton (2019) Soil Organic Matter Decomposition and Nitrogen Trace Gas Calculations of the DayCent Model. Technical report Natural Resource Ecology Laboratory, Colorado State University, Fort Collins, CO. Cited by: §A.1, §A.1, §A.1, §A.1, §A.1, §A.1, §A.1, §A.1, §2.2, §2.3. M. D. Hartman, W. J. Parton, S. J. D. Grosso, M. Easter, J. Hendryx, T. Hilinski, R. Kelly, C. A. Keough, K. Killian, S. Lutz, E. Marx, R. McKeown, S. Ogle, D. S. Ojima, K. Paustian, A. Swan, and S. Williams (2018) The Daily Century Ecosystem, Soil Organic Matter, Nutrient Cycling, Nitrogen Trace Gas, and Methane Model User Manual, Scientific Basis, and Technical Documentation. Technical report Natural Resource Ecology Laboratory, Colorado State University, Fort Collins, CO (en). External Links: Link Cited by: §A.1, §2.2. A. Hensen, U. Skiba, and D. Famulari (2013) Low cost and state of the art methods to measure nitrous oxide emissions. Environmental Research Letters 8 (2), p. 025022 (en). External Links: ISSN 1748-9326, Link, Document Cited by: §1. M. K. Jarecki, T. B. Parkin, A. S. K. Chan, J. L. Hatfield, and R. Jones (2008) Comparison of DAYCENT-Simulated and Measured Nitrous Oxide Emissions from a Corn Field. Journal of Environmental Quality 37 (5), p. 1685–1690 (en). Note: _eprint: https://acsess.onlinelibrary.wiley.com/doi/pdf/10.2134/jeq2007.0614 External Links: ISSN 1537-2537, Link, Document Cited by: §1. S. K. Jones, D. Famulari, C. F. Di Marco, E. Nemitz, U. M. Skiba, R. M. Rees, and M. A. Sutton (2011) Nitrous oxide emissions from managed grassland: a comparison of eddy covariance and static chamber measurements. Atmospheric Measurement Techniques 4 (10), p. 2179–2194 (English). External Links: ISSN 1867-1381, Link, Document Cited by: §1. D. R. Joshi, D. E. Clay, S. A. Clay, J. Moriles-Miller, A. L. M. Daigh, G. Reicks, and S. Westhoff (2024) Quantification and machine learning based N2O–N and CO2–C emissions predictions from a decomposing rye cover crop. Agronomy Journal 116 (3), p. 795–809 (en). Note: _eprint: https://acsess.onlinelibrary.wiley.com/doi/pdf/10.1002/agj2.21185 External Links: ISSN 1435-0645, Link, Document Cited by: §1. A. N. Kravchenko, E. R. Toosi, A. K. Guber, N. E. Ostrom, J. Yu, K. Azeem, M. L. Rivers, and G. P. Robertson (2017) Hotspots of soil N2O emission enhanced through water absorption by plant residue. Nature Geoscience 10 (7), p. 496–500 (en). External Links: ISSN 1752-0908, Link, Document Cited by: §1. S. S. Malhi and W. B. McGill (1981) Nitrification in three Alberta soils: Effect of temperature, moisture and substrate concentration. Soil Biology and Biochemistry 14 (en). External Links: Link Cited by: §A.1. M. Mastepanov (2026) Wirelessly controlled modular automatic chambers for greenhouse gas flux monitoring in natural and agricultural ecosystems. HardwareX 26, p. e00782. External Links: ISSN 2468-0672, Link, Document Cited by: §1. S. C. McClelland, K. Paustian, S. Williams, and M. E. Schipanski (2021) Modeling cover crop biomass production and related emissions to improve farm-scale decision-support tools. Agricultural Systems 191, p. 103151. External Links: ISSN 0308-521X, Link, Document Cited by: §1. R. J. Millington and R. C. Shearer (1971) Diffusion in aggregated porous media. Soil Science 111 (6), p. 372 (en-US). External Links: ISSN 0038-075X, Link Cited by: §A.1, §A.1. R. I. Papendick and G. S. Campbell (1981) Theory and Measurement of Water Potential. In Water Potential Relations in Soil Microbiology, Vol. 9 (SSSA Special Publications), p. 1–22 (en). External Links: Link, Document Cited by: §A.1, §A.1. T. B. Parkin (1987) Soil Microsites as a Source of Denitrification Variability. Soil Science Society of America Journal 51 (5), p. 1194–1199 (en). Note: _eprint: https://acsess.onlinelibrary.wiley.com/doi/pdf/10.2136/sssaj1987.03615995005100050019x External Links: ISSN 1435-0661, Link, Document Cited by: §1. W. J. Parton, E. A. Holland, S. J. Del Grosso, M. D. Hartman, R. E. Martin, A. R. Mosier, D. S. Ojima, and D. S. Schimel (2001) Generalized model for NO x and N2O emissions from soils. Journal of Geophysical Research: Atmospheres 106 (D15), p. 17403–17419 (en). Note: _eprint: https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2001JD900101 External Links: ISSN 2156-2202, Link, Document Cited by: §A.1, §A.1, §A.1, §2.2, §2.2. W. J. Parton, A. R. Mosier, D. S. Ojima, D. W. Valentine, D. S. Schimel, K. Weier, and A. E. Kulmala (1996) Generalized model for N2 and N2O production from nitrification and denitrification. Global Biogeochemical Cycles 10 (3), p. 401–412 (en). External Links: Link, Document Cited by: §A.1, §A.1, §A.1, §A.1, §A.1, §A.1, §2.2, §2.2. E. Pattey, G. C. Edwards, R. L. Desjardins, D. J. Pennock, W. Smith, B. Grant, and J. I. MacPherson (2007) Tools for quantifying N2O emissions from agroecosystems. Agricultural and Forest Meteorology 142 (2), p. 103–119. External Links: ISSN 0168-1923, Link, Document Cited by: §1. A. Philibert, C. Loyce, and D. Makowski (2013) Prediction of N2O emission from local information with Random Forest. Environmental Pollution 177, p. 156–163 (eng). External Links: ISSN 1873-6424, Document Cited by: §1. R. Phillips, D. W.T. Griffith, F. Dijkstra, G. Lugg, R. Lawrie, and B. Macdonald (2013) Tracking Short-Term Effects of Nitrogen-15 Addition on Nitrous Oxide Fluxes Using Fourier-Transform Infrared Spectroscopy. Journal of Environmental Quality 42 (5), p. 1327–1340 (en). Note: _eprint: https://acsess.onlinelibrary.wiley.com/doi/pdf/10.2134/jeq2013.02.0067 External Links: ISSN 1537-2537, Link, Document Cited by: §1. C. S. Potter, E. A. Davidson, and L. V. Verchot (1996) Estimation of global biogeochemical controls and seasonality in soil methane consumption. Chemosphere 32 (11), p. 2219–2246. External Links: ISSN 0045-6535, Link, Document Cited by: §A.1, §A.1, §A.1, §A.1, §A.1, §A.1, §A.1, §A.1. M. J. Prather and C. P. Wilson (2026) Projecting nitrous oxide over the 21st century, uncertainty related to stratospheric loss. Proceedings of the National Academy of Sciences 123 (6), p. e2524123123. External Links: Link, Document Cited by: §1. M. Raissi, P. Perdikaris, and G. E. Karniadakis (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, p. 686–707. External Links: ISSN 0021-9991, Link, Document Cited by: §1, §2.1, §2.1, §4. T. D. Rapson and H. Dacres (2014) Analytical techniques for measuring nitrous oxide. TrAC Trends in Analytical Chemistry 54, p. 65–74. External Links: ISSN 0165-9936, Link, Document Cited by: §1. A. R. Ravishankara, J. S. Daniel, and R. W. Portmann (2009) Nitrous Oxide (N2O): The Dominant Ozone-Depleting Substance Emitted in the 21st Century. Science 326 (5949), p. 123–125. External Links: Link, Document Cited by: §1. E. Ritchey, J. McGrath, and D. Gehring (2015) Determining Soil Texture by Feel. Agriculture and Natural Resources Publications. External Links: Link Cited by: §A.1. P. Robertson and D. Saha (2026) DSFAS project: an nitrous oxide (N2O) dataset to support data-driven modeling. KBS LTER. External Links: Link Cited by: §2.3, §2.3. P. Rochette, N. Tremblay, E. Fallon, D. A. Angers, M. H. Chantigny, J. D. MacDonald, N. Bertrand, and L.‐É. Parent (2010) N2O emissions from an irrigated and non‐irrigated organic soil in eastern Canada as influenced by N fertilizer addition. European Journal of Soil Science 61 (2), p. 186–196 (en). External Links: ISSN 1351-0754, 1365-2389, Link, Document Cited by: §1. D. Saha, B. Basso, and G. P. Robertson (2021) Machine learning improves predictions of agricultural nitrous oxide (N2O) emissions from intensively managed cropping systems. Environmental Research Letters 16 (2), p. 024004 (en). External Links: ISSN 1748-9326, Link, Document Cited by: §1. K. E. Saxton, W. J. Rawls, J. S. Romberger, and R. I. Papendick (1986a) Erratum: Estimating Generalized Soil-water Characteristics from Texture. Soil Science Society of America Journal 50 (4), p. NP–NP (en). Note: _eprint: https://acsess.onlinelibrary.wiley.com/doi/pdf/10.2136/sssaj1986.03615995005000040054x External Links: ISSN 1435-0661, Link, Document Cited by: §A.1. K. E. Saxton, W. J. Rawls, J. S. Romberger, and R. I. Papendick (1986b) Estimating Generalized Soil-water Characteristics from Texture. Soil Science Society of America Journal 50 (4), p. 1031–1036 (en). Note: _eprint: https://acsess.onlinelibrary.wiley.com/doi/pdf/10.2136/sssaj1986.03615995005000040039x External Links: ISSN 1435-0661, Link, Document Cited by: §A.1, §A.1. D. S. Schimel and W. J. Parton (1986) Microclimatic controls of nitrogen mineralization and nitrification in shortgrass steppe soils. Plant and Soil 93 (3), p. 347–357 (en). External Links: ISSN 1573-5036, Link, Document Cited by: §A.1. P. Sharma, B. Basso, A. Manuraj, M. S. Murillo, N. Millar, T. Tadiello, M. Sharma, M. Delandmeter, and G. P. Robertson (2026) Coupled machine learning-ecosystem ensemble models substantially improve predictions of nitrous oxide (N2O) fluxes from US croplands. Proceedings of the National Academy of Sciences of the United States of America 123 (10), p. e2524808123 (eng). External Links: ISSN 1091-6490, Link, Document Cited by: §1, §1, §3.2. H. Tian, N. Pan, R. L. Thompson, J. G. Canadell, P. Suntharalingam, P. Regnier, E. A. Davidson, M. Prather, P. Ciais, M. Muntean, S. Pan, W. Winiwarter, S. Zaehle, F. Zhou, R. B. Jackson, H. W. Bange, S. Berthet, Z. Bian, D. Bianchi, A. F. Bouwman, E. T. Buitenhuis, G. Dutton, M. Hu, A. Ito, A. K. Jain, A. Jeltsch-Thömmes, F. Joos, S. Kou-Giesbrecht, P. B. Krummel, X. Lan, A. Landolfi, R. Lauerwald, Y. Li, C. Lu, T. Maavara, M. Manizza, D. B. Millet, J. Mühle, P. K. Patra, G. P. Peters, X. Qin, P. Raymond, L. Resplandy, J. A. Rosentreter, H. Shi, Q. Sun, D. Tonina, F. N. Tubiello, G. R. van der Werf, N. Vuichard, J. Wang, K. C. Wells, L. M. Western, C. Wilson, J. Yang, Y. Yao, Y. You, and Q. Zhu (2024) Global nitrous oxide budget (1980–2020). Earth System Science Data 16 (6), p. 2543–2604 (English). External Links: ISSN 1866-3508, Link, Document Cited by: §1. H. Tian, R. Xu, J. G. Canadell, R. L. Thompson, W. Winiwarter, P. Suntharalingam, E. A. Davidson, P. Ciais, R. B. Jackson, G. Janssens-Maenhout, M. J. Prather, P. Regnier, N. Pan, S. Pan, G. P. Peters, H. Shi, F. N. Tubiello, S. Zaehle, F. Zhou, A. Arneth, G. Battaglia, S. Berthet, L. Bopp, A. F. Bouwman, E. T. Buitenhuis, J. Chang, M. P. Chipperfield, S. R. Dangal, E. Dlugokencky, J. W. Elkins, B. D. Eyre, B. Fu, B. Hall, A. Ito, F. Joos, P. B. Krummel, A. Landolfi, G. G. Laruelle, R. Lauerwald, W. Li, S. Lienert, T. Maavara, M. MacLeod, D. B. Millet, S. Olin, P. K. Patra, R. G. Prinn, P. A. Raymond, D. J. Ruiz, G. R. van der Werf, N. Vuichard, J. Wang, R. F. Weiss, K. C. Wells, C. Wilson, J. Yang, and Y. Yao (2020) A comprehensive quantification of global nitrous oxide sources and sinks.. Nature 586 (7828), p. 248–256 (en). External Links: ISSN 0028-0836, Link, Document Cited by: §1. P. A. Turner, T. J. Griffis, D. J. Mulla, J. M. Baker, and R. T. Venterea (2016) A geostatistical approach to identify and mitigate agricultural nitrous oxide emission hotspots. Science of The Total Environment 572, p. 442–449. External Links: ISSN 0048-9697, Link, Document Cited by: §1. N. Vemuri (2024) Developing a hybrid data-driven and informed model for prediction and mitigation of agricultural nitrous oxide flux hotspots. Frontiers in Environmental Science 12 (English). External Links: ISSN 2296-665X, Link, Document Cited by: §1. C. Wagner-Riddle, E. M. Baggs, T. J. Clough, K. Fuchs, and S. O. Petersen (2020) Mitigation of nitrous oxide emissions in the context of nitrogen loss reduction from agroecosystems: managing hot spots and hot moments. Current Opinion in Environmental Sustainability 47, p. 46–53. External Links: ISSN 1877-3435, Link, Document Cited by: §1. Y. Wang, I. A. Weeks, C. P. Meyer, and I. E. Galbally (2001) Two automatic chamber techniques for measuring soil-atmosphere exchanges of trace gases and results of their use in the OASIS field experiment. CSIRO Publishing, Aspendale Vic Australia (English). External Links: ISBN 0-643-06645-4, Link Cited by: §1. S. Yamulki, I. Wolf, R. Bol, B. Grant, R. Brumme, E. Veldkamp, and S. C. Jarvis (2000) Effects of dung and urine amendments on the isotopic content of N(2)O released from grasslands. Rapid communications in mass spectrometry: RCM 14 (15), p. 1356–1360 (eng). External Links: ISSN 0951-4198, Document Cited by: §1. Z. Zhao, X. Ding, and B. A. Prakash (2024) PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks. arXiv. Note: arXiv:2307.11833 [cs.CE] External Links: Link, Document Cited by: §4. Appendix A Technical appendices and supplementary material A.1 Auxiliary Equations Following Hartman and Parton (2019), the equations for the respective effects of soil temperature, water-filled pore space, and soil pH on nitrification are given below. Fn(Ts)F_n(T_s) is computed via a generalized Poisson density function following Hartman and Parton (2019) (Equations 3.13–3.14), parameterized by the long-term maximum monthly air temperature (Ta,maxT_a,max). Fn(Ts)=(A1−TsA1−Ta,max)A2⋅exp(A2A3(1.0−(A1−TsA1−Ta,max)A3))if Ta,max≥35.0f_gen_poisson_density(Ts+(A0−Ta,max),A0,A1,A2,A3)if Ta,max<35.0F_n(T_s)= cases ( A_1-T_sA_1-T_a,max )^A_2· ( A_2A_3 (1.0- ( A_1-T_sA_1-T_a,max )^A_3 ) )&if T_a,max≥ 35.0\\[10.0pt] f\_gen\_poisson\_density (T_s+(A_0-T_a,max),\ A_0,\ A_1,\ A_2,\ A_3 )&if T_a,max<35.0 cases (10) Variable Description Units Source Fn(Ts)F_n(T_s) effect of soil temperature on nitrification unitless TsT_s average soil temperature of soil layers 2 and 3 ∘C Cycles A0A_0 peak location parameter (bell curve center) ∘C constant at 35.0 A1A_1 lower bound parameter ∘C constant at -5.0 A2A_2 spread shape parameter unitless constant at 4.5 A3A_3 asymmetry shape parameter unitless constant at 7.0 Hartman and Parton (2019) refers to Ta,maxT_a,max somewhat ambiguously as the "long-term maximum monthly air temperature", which could be interpreted to mean that Ta,maxT_a,max is a site-specific value, calculated by taking the average temperature across each month for a site, then finding the maximum across all of them. However, Parton et al. (2001) states on Page 17,406: "The effect of soil temperature on nitrification is based on data presented by Malhi and McGill (1981) which show that there is an optimal temperature for nitrification that is a function of the average maximum monthly air temperature for the warmest month of the year." This statement could also be reasonably interpreted to mean that the "average maximum monthly air temperature for the warmest month of the year" refers to Ta,maxT_a,max, such that Ta,maxT_a,max is calculated by finding the warmest month of the year, extracting that month’s average temperature. In summary, there are essentially three possible different interpretations of how to define Ta,maxT_a,max, each of which are defensible in their own way: (1) maximum of monthly mean of average daily temperature TavgT_avg, (2) maximum of monthly mean of maximum daily temperature TmaxT_max, and (3) mean of daily maximum temperature TmaxT_max in the warmest month. In the face of these two ambiguous and somewhat conflicting variable descriptions from Hartman and Parton (2019) and Parton et al. (2001), we choose to define "maximum monthly air temperature" as the monthly mean maximum temperature, and move forward with Interpretation 2. This is supported by Section 3.6.1 of the DayCent manual, which defines the variable TMX2M (1-12) ∘C as the "mean monthly maximum air temperature" (Hartman et al., 2018). We therefore define Ta,maxT_a,max as the maximum of TMX2M across all months. Although neither Hartman and Parton (2019) nor Parton et al. (2001) explicitly offer a formula for Ta,maxT_a,max, our interpretation as described above allows us to establish the following equation as our definition of Ta,maxT_a,max. Ta,max=maxM∈1,…,12(Ta,M)T_a, = _M∈\1,…,12\ (T_a,M ) (11) Variable Description Units Source Ta,maxT_a,max long-term maximum monthly air temperature ∘C Ta,MT_a,M monthly mean maximum temperature ∘C Cycles Fn(WFPS)F_n(WFPS) is computed via Fn(WFPS)=1.01.0+30.0⋅exp(−9.0⋅wcrel)if wcrel≤1.0−1.01.0−θfc⋅(WFPS−1.0)if wcrel>1.0F_n(WFPS)= cases 1.01.0+30.0· (-9.0· wc_rel)&if wc_rel≤ 1.0\\[10.0pt] -1.01.0- _fc·(WFPS-1.0)&if wc_rel>1.0 cases (12) Variable Description Units Source Fn(WFPS)F_n(WFPS) effect of water-filled pore space on nitrification unitless wcrelwc_rel weighted average relative water content in the 2nd and 3rd soil layers unitless Equation 13a θfc _fc volumetric water content at field capacity unitless Equation 37a WFPSWFPS weighted average water-filled pore space in 2nd and 3rd soil layers unitless Equation 16a The upcoming subequations follow from Hartman and Parton (2019), under the definition of wcrelwc_rel used herein, where θfc,l _fc,l denotes θfc _fc evaluated for layer l using that layer’s sand and clay fractions. Note that widthlwidth_l can be found in the "soilLayersCN.txt" Cycles output file, but is technically a "Cycles configuration" (as opposed to a typical Cycles-simulated value) since there is no universal Cycles convention for layer widths and labeling. Correspondingly, note that Hartman and Parton (2019) did not provide an explicit convention for "2nd and 3rd soil layers", so we choose to let the 2nd and 3rd layers refer to the depths of 5 cm to 10 cm, and 10 to 20 cm, below the surface, respectively. wcrel wc_rel =∑l=23widthl⋅wcrel,l∑l=23widthl = _l=2^3width_l· wc_rel,l _l=2^3width_l (13a) wcrel,l wc_rel,l =swclwidthl−swclimitlθfc,l−swclimitl = swc_lwidth_l-swclimit_l _fc,l-swclimit_l (13b) Variable Description Units Source wcrelwc_rel weighted average relative water content in the 2nd and 3rd soil layers unitless wcrel,lwc_rel,l relative water content of soil layer l unitless l specific soil layer index unitless constant at 2 or 3 widthlwidth_l thickness of soil layer l cm Cycles swclswc_l soil water content of soil layer l cm H2O Equation 14 swclimitlswclimit_l minimum volumetric soil water content (wilting point) of layer l unitless Equation 15 θfc,l _fc,l volumetric soil water content of layer l at field capacity unitless Equation 37a Since Cycles only provides volumetric water content, we derive swclswc_l via the standard conversion formula from volumetric water content to equivalent water depth: swcl=VWCl⋅widthlswc_l=VWC_l· width_l (14) Variable Description Units Source swclswc_l soil water content of soil layer l cm H2O VWClVWC_l volumetric water content of soil layer l m3 m-3 Cycles widthlwidth_l thickness of soil layer l cm Cycles We derive swclimitlswclimit_l from Equation 2 of Saxton et al. (1986b)’s pedotransfer functions, for when the applied water tension reaches 1500 kPa; resulting in the equation Ψ=1500=AsxswclimitlBsx =1500=A_sxswclimit_l^B_sx, which can be rearranged into the form swclimitl=(1500Asx)1Bsxswclimit_l= ( 1500A_sx ) 1B_sx (15) Variable Description Units Source swclimitlswclimit_l minimum volumetric soil water content (wilting point) of layer l unitless AsxA_sx air entry potential scaling parameter kPa Equation 37b BsxB_sx pore size distribution shape parameter unitless Equation 37c Similarly, the following two subequations come from Hartman and Parton (2019) and from the definition of WFPSWFPS used herein. Note that PDPD’s value is an assumption from Equation 27. WFPS WFPS =∑l=23widthl⋅WFPSl∑l=23widthl = _l=2^3width_l· WFPS_l _l=2^3width_l (16a) WFPSl WFPS_l =swclwidthl1.0−BDlPD = swc_lwidth_l1.0- BD_lPD (16b) Variable Description Units Source WFPSWFPS weighted average water-filled pore space in 2nd and 3rd soil layers unitless WFPSlWFPS_l water-filled pore space of specific soil layer l unitless l specific soil layer index unitless constant at 2 or 3 widthlwidth_l thickness of the layer cm Cycles swclswc_l soil water content of soil layer l cm H2O Equation 14 BDlBD_l soil bulk density of soil layer l g cm-3 Cycles PDPD particle density g cm-3 constant at 2.65 Fn(pH)F_n(pH) is computed via Fn(pH)=B1+B2π⋅arctan(π⋅B3⋅(pH−B0))F_n(pH)=B_1+ B_2π· (π· B_3·(pH-B_0) ) (17) Variable Description Units Source Fn(pH)F_n(pH) effect of soil pH on nitrification unitless pHpH soil pH in the 2nd soil layer unitless static site measurement B0B_0 inflection point parameter unitless constant at 5.0 B1B_1 vertical offset parameter unitless constant at 0.56 B2B_2 amplitude parameter unitless constant at 1.0 B3B_3 sigmoid transition steepness parameter unitless constant at 0.45 The following equation comes from Parton et al. (1996). Note that we calculate NO3NO_3 as the weighted average across layers 2 and 3, as per Hartman and Parton (2019)’s convention (where the weights are the soil layer widths). ϕmax(NO3)=11,000+40,000⋅arctan(π⋅0.002⋅(NO3−180))π _max(NO_3)=11,000+ 40,000· (π· 0.002·(NO_3-180))π (18) Variable Description Units Source ϕmax(NO3) _max(NO_3) max ϕN,den _N,den for a given NO3 level, assuming high CO2 g N ha-1 d-1 NO3NO_3 soil NO3 concentration mg N kg-1 Cycles ϕmax(CO2)=24,0001+200e0.35⋅CO2−100 _max(CO_2)= 24,0001+ 200e^0.35· CO_2-100 (19) Variable Description Units Source ϕmax(CO2) _max(CO_2) max ϕN,den _N,den for a given respiration rate, assuming high NO3 g N ha-1 d-1 CO2CO_2 soil heterotrophic respiration rate (excluding root respiration) kg C ha-1 d-1 Equation 20 Note that in Equation 20, we follow Parton et al. (1996) in assuming that FCO2(WFPS)=Fn(WFPS)F_CO_2(WFPS)=F_n(WFPS) and that FCO2(Ts)=Fn(Ts)F_CO_2(T_s)=F_n(T_s). This follows from Parton et al. (1996)’s statement in the last paragraph of page 407: "SwS_w and StS_t are the same functions shown in Figures 2a and 2b, respectively", where Figures 2a and 2b represent Fn(WFPS)F_n(WFPS) and Fn(Ts)F_n(T_s), respectively. CO2=CO2,max⋅FCO2(WFPS)⋅FCO2(Ts)CO_2=CO_2,max· F_CO_2(WFPS)· F_CO_2(T_s) (20) Variable Description Units Source CO2CO_2 soil heterotrophic respiration rate (excluding root respiration) kg C ha-1 d-1 CO2,maxCO_2,max maximum soil respiration rate kg C ha-1 d-1 constant at 80 FCO2(WFPS)F_CO_2(WFPS) effect of water-filled pore space on soil respiration rate unitless Equation 12 FCO2(Ts)F_CO_2(T_s) effect of soil temperature on soil respiration rate unitless Equation 10 Continuing from Parton et al. (1996): Fd(WFPS)=ab(cb(d⋅WFPS))F_d(WFPS)= ab ( cb^(d· WFPS) ) (21) Variable Description Units Source Fd(WFPS)F_d(WFPS) effect of WFPS on denitrification unitless a amplitude parameter unitless Table 3 b threshold parameter unitless Table 3 c steepness parameter unitless Table 3 d curvature parameter unitless Table 3 The following table comes from Parton et al. (1996): Table 3: Parameters for Fd(WFPS)F_d(WFPS) in Equation 21 based on soil texture Parameter Soil Texture a b c d sandy 1.56 12.0 16.0 2.01 medium 4.82 14.0 16.0 1.39 fine 60.0 18.0 22.0 1.06 Unfortunately, Parton et al. (1996) does not explicitly provide a soil texture classification method, i.e., how soil and clay percentages translate into "sandy", "medium", or "fine". As such, we classify soil texture following the USDA broad texture family groupings (coarse/medium/fine) based on sand and clay fractions from the Cycles model configuration, where "coarse" is equivalent to "sandy", "loamy" is equivalent to "medium", and "clayey" is equivalent to "fine". The USDA texture classification system uses 12 different categories, so we follow Ritchey et al. (2015)’s convention, in which: 1. Sandy soil is soil with 70% to 100% sand (this category consists of USDA categories "sand" and "loamy sand") 2. Fine soil is soil with 35% to 40% clay (this category consists of USDA categories "sandy clay", "clay", and "silty clay") 3. Medium soil is all other soil that does not fall into the "sandy" or "fine" category (this category consists of USDA categories "sandy clay loam", "sandy loam", "clay loam", "loam", "silty clay loam", "silt loam", and "silt") Following Del Grosso et al. (2000): Fr(NO3/CO2)=max(0.16⋅k1,k1⋅e−0.8(NO3CO2,μ))F_r(NO_3/CO_2)= (0.16· k_1,\ k_1· e^-0.8 ( NO_3CO_2,μ )) (22) Variable Description Units Source Fr(NO3/CO2)F_r(NO_3/CO_2) NO3-to-respiration ratio factor unitless NO3NO_3 soil NO3 concentration μ N gsoil-1 Cycles CO2,μCO_2,μ heterotrophic CO2 respiration μ C gsoil-1 d-1 Equation 23 k1k_1 maximum N2/N2O ratio (asymptotic ceiling) unitless Equation 24 Note that NO3NO_3’s units of "mg N kg-1" in Equation 18 and "μ N gsoil-1" in Equation 22 are equivalent. However, CO2,μCO_2,μ from Del Grosso et al. (2000) is not the same as CO2CO_2 from Parton et al. (1996); CO2,μCO_2,μ (which is measured in μ C gsoil-1 d-1) must be derived from CO2CO_2 (which is measured in kg C ha-1 d-1) via the following conversion formula, which relies on bulk density. We follow Del Grosso et al. (2000) in assuming an active soil depth z of 20 cm, due to Schimel and Parton (1986)’s observation that most N cycling occurs in the top 5 cm of mineral soil. As Del Grosso et al. (2000) states verbatim on page 1056: "⋯·s an active soil depth of 20 cm is assumed because the majority of N cycling occurs in the top 5 cm of mineral soil and N cycling decreases dramatically with depth (Schimel and Parton (1986))." CO2,μ=CO210⋅BD⋅zCO_2,μ= CO_210· BD· z (23) Variable Description Units Source CO2,μCO_2,μ heterotrophic CO2 respiration μ C gsoil-1 d-1 CO2CO_2 soil heterotrophic respiration rate (excluding root respiration) kg C ha-1 d-1 Equation 20 BDBD bulk density g cm-3 Cycles z soil layer depth m constant at 0.20 Continuing from Del Grosso et al. (2000): k1=max(1.7, 38.4−350⋅DFC)k_1= (1.7,\ 38.4-350· D_FC) (24) Variable Description Units Source k1k_1 maximum N2/N2O ratio (asymptotic ceiling) unitless DFCD_FC gas diffusivity at field capacity unitless Equation 38 In order to obtain DFCD_FC, we must begin by finding DsD0 D_sD_0. Following Potter et al. (1996) as in Del Grosso et al. (2000), we use the formulation of Millington and Shearer (1971) with polynomial approximations of the exponent terms: DsD0=(1−SWA)2⋅[A−ΘA+S]2z⋅[1−P2x]⋅[(P−ΘP)−(P−ΘP)2y](1−SWA)2⋅[A−ΘA+S]2⋅[1−P2x]+(P−ΘP)−(P−ΘP)2y+(1−SWP)2(P−ΘP)2y D_sD_0= (1-S_WA)^2·[A- _AA+S]^2z·[1-P^2x]·[(P- _P)-(P- _P)^2y](1-S_WA)^2·[A- _AA+S]^2·[1-P^2x]+(P- _P)-(P- _P)^2y+(1-S_WP)^2(P- _P)^2y (25) Variable Description Units Source DsD0 D_sD_0 normalized soil gas diffusivity at field capacity unitless S solid phase volume fraction m3 m-3 Equation 26 A intra-aggregate pore space m3 m-3 Equation 28 P inter-aggregate pore space m3 m-3 Equation 29 ΘA _A volumetric water content in intra-aggregate pores at FC m3 m-3 Equation 34 ΘP _P volumetric water content in inter-aggregate pores at FC m3 m-3 Equation 31 SWAS_WA fractional liquid saturation of intra-aggregate pore space unitless Equation 30, 32 SWPS_WP fractional liquid saturation of inter-aggregate pore space unitless Equation 33, 35 x tortuosity exponent for inter-aggregate pore space unitless Equation 36a y tortuosity exponent for inter-aggregate water content unitless Equation 36b z tortuosity exponent for intra-aggregate pore space unitless Equation 36c Note that in Equation 25, we follow the literal typesetting of Potter et al. (1996)’s equation, treating ΘA _AA as a standalone fraction, rather than treating A−ΘA+S A- _AA+S as a fraction. That said, the ambiguity doesn’t affect the results of our paper, since the term (1−SWA)2(1-S_WA)^2 where SWA=1S_WA=1 simplifies the entire multiplicative expression to 0 anyway. Following Potter et al. (1996)’s estimate: S=1−PCS=1-PC (26) Variable Description Units Source S solid phase volume fraction m3 m-3 PCPC total pore space capacity (porosity) m3 m-3 Equation 27 To find PCPC, we use Davidson and Trumbore (1995)’s assumed particle density value: PC=1−BDPDPC=1- BDPD (27) Variable Description Units Source PCPC total pore space capacity (porosity) m3 m-3 BDBD bulk density g cm-3 Cycles PDPD particle density g cm-3 constant at 2.65 To find A, we follow Davidson and Trumbore (1995)’s lead in equating A with θfc _fc: A=θfcA= _fc (28) Variable Description Units Source A intra-aggregate pore space m3 m-3 θfc _fc volumetric water content at field capacity m3 m-3 Equation 37a We similarly follow Potter et al. (1996) in estimating P: P=|PC−A|P=|PC-A| (29) Variable Description Units Source P inter-aggregate pore space m3 m-3 PCPC total pore space capacity (porosity) m3 m-3 Equation 27 A intra-aggregate pore space m3 m-3 Equation 28 To find ΘA _A and ΘP _P, we begin with Davidson and Trumbore (1995)’s assumption: "Intra-aggregate porosity was estimated from volumetric soil water content at field capacity …water has drained from the inter-aggregate pore spaces, leaving the intra-aggregate pore spaces 100% water-filled …it is assumed that the intra-aggregate spaces store water first and lose water last", which forms the basis of the following two equations: SWA=1S_WA=1 (30) ΘP=0 _P=0 (31) We also know from Potter et al. (1996) that "SWAS_WA and SWPS_WP denote the fractional liquid saturation of their respective A and P components of the total void volume", which forms the basis for the following two equations: SWA=ΘAAS_WA= _AA (32) Variable Description Units Source SWAS_WA fractional liquid saturation of intra-aggregate pore space unitless ΘA _A volumetric water content in intra-aggregate pores at field capacity m3 m-3 Equation 34 A intra-aggregate pore space m3 m-3 Equation 28 SWP=ΘPPS_WP= _PP (33) Variable Description Units Source SWPS_WP fractional liquid saturation of inter-aggregate pore space unitless ΘP _P volumetric water content in inter-aggregate pores at FC m3 m-3 Equation 31 P inter-aggregate pore space m3 m-3 Equation 29 By combining Equation 30 and Equation 32, we find that SWA=ΘA=1S_WA= _AA=1; or equivalently, when combined with Equation 28: ΘA=A=θfc _A=A= _fc (34) Variable Description Units Source ΘA _A volumetric water content in intra-aggregate pores at field capacity m3 m-3 A intra-aggregate pore space m3 m-3 Equation 28 By combining Equation 33 with Equation 31, we furthermore find: SWP=0S_WP=0 (35) Variable Description Units Source SWPS_WP fractional liquid saturation of inter-aggregate pore space unitless Finally, rather than solving numerically for x, y, and z as described by Millington and Shearer (1971), we will use Potter et al. (1996)’s polynomial approximations: x x =0.477P3−0.596P2+0.437P+0.564 =0.477P^3-0.596P^2+0.437P+0.564 (36a) y y =0.477(P−ΘP)3−0.596(P−ΘP)2+0.437(P−ΘP)+0.564 =0.477(P- _P)^3-0.596(P- _P)^2+0.437(P- _P)+0.564 (36b) z z =0.477[A−ΘA+S]3−0.596[A−ΘA+S]2+0.437[A−ΘA+S]+0.564 =0.477 [ A- _AA+S ]^3-0.596 [ A- _AA+S ]^2+0.437 [ A- _AA+S ]+0.564 (36c) Variable Description Units Source x tortuosity exponent for inter-aggregate pore space unitless y tortuosity exponent for inter-aggregate water content unitless z tortuosity exponent for intra-aggregate pore space unitless ΘA _A volumetric water content in intra-aggregate pores at FC m3 m-3 Equation 34 ΘP _P volumetric water content in inter-aggregate pores at FC m3 m-3 Equation 31 S solid phase volume fraction m3 m-3 Equation 26 A intra-aggregate pore space m3 m-3 Equation 28 P inter-aggregate pore space m3 m-3 Equation 29 Following Potter et al. (1996), we use Equation 2 from Saxton et al. (1986b) and assume the water potential (i.e., tension) to be 33 kPa for medium to fine textures, and 10 kPa for coarse textures, as in Papendick and Campbell (1981). (Note that Table 2 in Saxton et al. (1986b) has a known error; as noted in the published erratum Saxton et al. (1986a), Equation 37c should not have a standalone term of gsx(%S)2g_sx(\%S)^2.) This results in the equation Ψ=Asx(θfc)Bsx =A_sx( _fc)^B_sx, which will then rearrange into the following form, which nicely parallels Equation 15. Note that soil texture inputs (%S, %C) used throughout this section were derived from gSSURGO, SoilGrids, or original site-level publications depending on data availability at each site, as provided by the Cycles model developers. θfc _fc =(ΨAsx)1Bsx = ( A_sx ) 1B_sx (37a) Asx A_sx =100⋅exp(asx+bsx(%C)+csx(%S)2+dsx(%S)2(%C)) =100· (a_sx+b_sx(\%C)+c_sx(\%S)^2+d_sx(\%S)^2(\%C) ) (37b) Bsx B_sx =esx+fsx(%C)2+gsx(%S)2(%C) =e_sx+f_sx(\%C)^2+g_sx(\%S)^2(\%C) (37c) Variable Description Units Source θfc _fc volumetric water content at field capacity m3 m-3 AsxA_sx air entry potential scaling parameter kPa BsxB_sx pore size distribution shape parameter unitless Ψ water potential (tension) kPa Table 4 %S\%S percentage of soil that is sand unitless site survey %C\%C percentage of soil that is clay unitless site survey asxa_sx regression coefficient unitless constant at -4.396 bsxb_sx regression coefficient unitless constant at -0.0715 csxc_sx regression coefficient unitless constant at -4.880 × 10-4 dsxd_sx regression coefficient unitless constant at -4.285 × 10-5 esxe_sx regression coefficient unitless constant at -3.140 fsxf_sx regression coefficient unitless constant at -2.22 × 10-3 gsxg_sx regression coefficient unitless constant at -3.484 × 10-5 The following table comes from Potter et al. (1996)’s assumptions based on Papendick and Campbell (1981): Table 4: Parameters for Ψ in Equation 37a based on soil texture Soil Texture Water Potential Ψ (kPa) sandy 10.0 medium 33.0 fine 33.0 Finally, DFCD_FC is obtained by evaluating Equation 25 at θ=θfcθ= _fc, following the assumption of Davidson and Trumbore (1995) that intra-aggregate pore spaces are fully water-filled at field capacity (ΘA=θfc _A= _fc, SWA=1S_WA=1) and inter-aggregate pore spaces are air-filled (ΘP=0 _P=0, SWP=0S_WP=0): DFC=DsD0|ΘA=θfc,SWA=1,ΘP=0,SWP=0D_FC= . D_sD_0 |_ _A= _fc,S_WA=1, _P=0,S_WP=0 (38) Then, continued from Del Grosso et al. (2000), we use Hartman and Parton (2019)’s convention of treating WFPSWFPS as the average across the 2nd and 3rd soil layers specifically. Note that we multiply WFPSWFPS by 100; this doesn’t appear in the original equation from Del Grosso et al. (2000), but is necessary for us since we define WFPSWFPS as a fraction, whereas WFPSWFPS is treated as a percent in the original equation. Fr(WFPS)=max(0.1,0.015(100⋅WFPS)−0.32)if soil is intactmax(0.1,0.05(100⋅WFPS)−3.1)if soil is repackedF_r(WFPS)= cases (0.1,0.015(100· WFPS)-0.32)&if soil is intact\\[10.0pt] (0.1,0.05(100· WFPS)-3.1)&if soil is repacked cases (39) Variable Description Units Source Fr(WFPS)F_r(WFPS) effect of water-filled pore space on RN2/N2OR_N_2/N_2O unitless WFPSWFPS weighted average water-filled pore space in 2nd and 3rd soil layers unitless Equation 16a As explained in Section 2.3, we determined site-specific soil intactness designations (intact vs. repacked) from the historical tillage records available for each of our sites. A.2 Additional Figures Figure 4: Site 1 complete time series Figure 5: Site 2 complete time series Figure 6: Site 3 complete time series Figure 7: Site 4 complete time series A.3 Additional Tables Table 5: R2 across λ sweep (holdout validation) lambda 0 0.01 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 seed 41 0.442 0.479 0.439 0.449 0.358 0.322 0.274 0.227 0.166 0.16 0.19 0.173 0.135 42 0.44 0.438 0.435 0.432 0.402 0.376 0.358 0.306 0.3 0.275 0.242 0.235 0.188 43 0.379 0.338 0.391 0.346 0.28 0.233 0.168 0.133 0.019 -0.043 -0.06 -0.182 -0.224 44 0.412 0.421 0.386 0.369 0.335 0.291 0.187 0.159 0.14 0.106 0.112 0.091 0.113 45 0.45 0.457 0.434 0.459 0.419 0.402 0.375 0.364 0.299 0.244 0.234 0.25 0.189 46 0.395 0.399 0.399 0.402 0.382 0.368 0.343 0.312 0.263 0.254 0.161 0.191 0.168 47 0.386 0.354 0.318 0.344 0.284 0.24 0.191 0.161 0.115 0.115 0.081 0.052 0.055 48 0.399 0.396 0.387 0.374 0.359 0.312 0.267 0.22 0.169 0.11 0.093 0.074 0.037 49 0.43 0.439 0.434 0.431 0.409 0.369 0.343 0.288 0.258 0.219 0.192 0.153 0.099 50 0.374 0.343 0.338 0.351 0.299 0.25 0.23 0.156 0.155 0.148 0.144 0.139 0.145 mean 0.411 0.406 0.396 0.396 0.353 0.316 0.274 0.233 0.188 0.159 0.139 0.118 0.09 std 0.027 0.047 0.04 0.042 0.049 0.058 0.074 0.077 0.086 0.09 0.084 0.118 0.116 Table 6: R2 across λ sweep (leave-one-site-out validation) lambda 0 0.01 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 seed 41 -12.466 -7.126 -60.352 -16.591 -19.251 -11.576 -16.524 -3.431 -6.179 -4 -4.233 -3.729 -5.936 42 -18.642 -25.336 -30.724 -7.637 -23.205 -32.59 -7.347 -9.48 -2.452 -3.247 -4.539 -3.769 -6.531 43 -180.562 -78.535 -29.245 -1.539 -27.242 -203.175 -129.547 -80.989 -101.11 -21.849 -56.527 -37.617 -47.983 44 -11.944 -39.53 -17.288 -28.272 -45.996 -11.974 -3.589 -3.814 -3.265 -16.556 -4.855 -4.87 -5.273 45 -85.602 -62.964 -33.413 -26.55 -87.985 -45.606 -5.258 -31.819 -24.96 -32.353 -34.366 -4.598 -41.846 46 -48.069 -0.883 -1.529 -115.415 -77.687 -12.285 -2.369 -13.605 -23.389 -3.294 -3.705 -5.343 -7.065 47 -74.384 -118.931 -32.469 -65.23 -47.697 -102.69 -60.357 -22.196 -39.976 -30.938 -42.735 -53.996 -40.265 48 -63.324 -95.2 -77.864 -42.524 -5.018 -19.809 -19.711 -25.817 -8.816 -4.218 -5.629 -5.091 -6.625 49 -147.117 -311.641 -136.065 -101.067 -109.851 -30.654 -19.079 -6.473 -4.287 -6.084 -3.447 -4.956 -4.214 50 -74.608 -6.746 -92.948 -33.315 -46.036 -31.805 -10.782 -6.218 -4.532 -3.43 -1.873 -5.343 -6.353 mean -71.672 -74.689 -51.190 -43.814 -48.997 -50.216 -27.456 -20.384 -21.897 -12.597 -16.191 -12.931 -17.209 std 53.170 87.736 38.633 36.540 31.660 57.028 37.588 22.266 28.928 11.281 19.246 16.849 17.236