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S$^{2}$-PINN: Stochastic Separable Physics-Informed Neural Networks

Zhendong Li, Akwum Onwunta

arXiv:2610.03303v1cs.LGmath-phmath.MP

TL;DR

Random-PDE UQ needs tractable moments and calibration, while existing neural methods often do not exploit the stochastic structure needed for them. S2-PINN combines separated spatial, temporal, and stochastic representations with physics and projected-residual constraints. The paper reports improved moment accuracy and calibration across its evaluations, while noting limits from finite rank and stochastic truncation.

  • Problem

    Existing neural solvers often ignore stochastic structure, while classical spectral approaches face dimensional costs in random-PDE uncertainty quantification.

    Source: §1 · p. 2

  • Method

    S2-PINN couples learnable Gaussian spatial features, Fourier temporal features, and gPC stochastic modes through a low-rank CP representation trained with strong-form and projected residual losses.

    Source: §1 · p. 2, §3.1 · p. 4, §3.2 · p. 4

  • Results

    Across manufactured and additional evaluations, S2-PINN improves moment accuracy and calibration; on four manufactured PDEs its coverage closely follows nominal levels while MC-dropout under-covers.

    Source: §1 · p. 2, §4.3 · p. 7

  • Takeaways & Limitations

    The results support stochastic separability as an effective design principle, especially when solutions have low-rank space–time–random structure and a useful low-order stochastic expansion.

    Source: §5 · p. 10

    The theory does not provide a finite-q, finite-G end-to-end solution-error guarantee, and finite rank and stochastic truncation introduce approximation errors.

    Source: §5 · p. 10

Abstract

from arXiv · show

Uncertainty quantification (UQ) for random partial differential equations (PDEs) is ubiquitous in computational science and engineering. However, classical spectral solvers for this class of problems face the curse of dimensionality, and existing neural solvers often ignore the stochastic structure that makes moments and calibration tractable. We introduce a stochastic separable physics-informed neural network, dubbed S$^{2}$-PINN, that represents the solution $u(t,\mathbf{x},\mathbf{Z})$ of a random PDE with a learnable Gaussian spatial dictionary, Fourier temporal features, and a generalized polynomial chaos (gPC) stochastic basis, coupled by a low-rank Canonical Polyadic (CP) tensor decomposition core. The method is trained with a hybrid strong-form and gPC-projected residual loss. Our theoretical analysis establishes that the separable class is dense in $L^2$ under mild conditions, and the projected residual corresponds exactly to a stochastic Galerkin constraint. Furthermore, we show that mini-batch projection coefficients are logarithmically dependent on the number of gPC modes, and that the orthogonality penalty controls the conditioning of the learned spatial dictionary. Using four manufactured random PDE benchmarks, we show that S$^{2}$-PINN outperforms nine baselines in terms of mean and variance accuracy, as well as calibration, while using significantly fewer parameters. Further evaluations on non-manufactured Poisson and Darcy problems, a stochastic Navier--Stokes problem, a diffusion scaling study of higher random dimensions, and two stochastic inverse problems reveal the generalization capabilities of the proposed structure. Together, these results support stochastic separability as an effective design principle for physics-informed neural UQ. The code for the experiments can be found in https://github.com/DMax1314/s2pinn

1 Introduction

Random-PDE UQ needs tractable moments and calibration, but existing neural architectures do not jointly exploit separability across space, time, and randomness. S2-PINN combines those structures with physics and stochastic-projection constraints, and the paper reports improved moment accuracy and calibration.

Motivation

  • Existing neural UQ architectures do not jointly exploit separability across space, time, and randomness, leaving an integrated stochastic PINN with physics and projection constraints as a missing ingredient.Classical Monte Carlo, stochastic Galerkin, and collocation solvers are reliable but costly because they require repeated solves or large coupled systems.Source: §1 · pp. 1–2

Contributions

  • S2-PINN couples a learnable spatial dictionary, temporal Fourier features, and gPC modes through a rank-q CP factorization, with strong-form and projected residual losses.The coupling complexity is O((M+Mt+G)q), rather than dense O(MMtG).Source: §1 · p. 2
  • The analysis establishes separable expressivity, projection–Galerkin equivalence, O(log G) mini-batch concentration, and orthogonality-based conditioning control.The projected residual enforces stochastic Galerkin consistency alongside local PDE consistency.Source: §1 · p. 2

Results and scope

  • The paper reports improved moment accuracy and calibration, with scaling and inverse-problem studies extending evaluation to higher-dimensional random inputs and joint recovery of fields and parameters.The central claim is strongest when solutions have low-rank space–time–random structure and a useful low-order stochastic expansion.Source: §1 · p. 2

2 Related Work

Prior UQ solvers and neural methods leave a gap between structural compression, explicit stochastic modeling, and physics constraints. S2-PINN combines these elements in a physics-informed, low-rank representation.

Physics-Informed and Operator Learning for UQ

  • Existing UQ approaches use labeled data, dense stochastic representations, or fail to exploit multidimensional structural separation, motivating a physics-informed architecture that isolates spatial derivative costs from stochastic basis size.The paper also compares against PC2, which constrains a fixed global polynomial surrogate through KKT-style regression.Source: §2 · p. 2

Low-Rank Tensor Methods in Neural Solvers

  • Neural-tensor hybrids compress weights or factorize spatial axes but lack an explicit stochastic polynomial basis and UQ-specific Galerkin projection.S2-PINN instead combines a learnable spatial dictionary, separated gPC modes, and hybrid strong-form and projection constraints.Source: §2 · p. 3

Separable PINNs for High-Dimensional PDEs

  • Adding stochastic coordinates to SPINN naively creates a dense coefficient tensor that grows multiplicatively with spatial, temporal, and stochastic resolutions; S2-PINN uses low-rank CP coupling with gPC modes.The paper compares with SPINN+Z, functional CP-PINNs, and Neural Chaos to isolate this design choice.Source: §2 · p. 3

3 Method

S2-PINN assumes a finite random vector with known marginals and represents random-PDE solutions through separated spatial, temporal, and stochastic factors. Its hybrid physics loss combines pointwise residual constraints with stochastic projection, supported by theoretical analysis of the representation and regularization.

3 Method

  • The method assumes uncertainty is encoded by a finite random vector with known marginals, enabling an orthonormal gPC basis and a low-rank CP coupling of spatial, temporal, and stochastic factors.The loss combines pointwise physics constraints with a stochastic projection constraint.Source: §3 · p. 3

3.1 Three-way Separable Representation

  • A rank-q CP expansion couples Gaussian spatial atoms, learnable Fourier temporal features, and gPC stochastic modes, with arithmetic scaling as O(Bq(M+Mt+G)) rather than O(BMMtG).Under density assumptions on the feature families, the union of rank-q separable functions is dense in L2.Source: §3.1 · pp. 3–4

3.2 Hybrid Physics Loss and Regularization

  • The hybrid objective minimizes the strong-form PDE residual alongside projected residual and boundary/initial-condition terms, while spatial Gram regularization encourages well-conditioned features.The projected term is Galerkin-consistent and statistically stable, but can introduce bias when important stochastic energy lies outside the chosen polynomial space.Source: §3.2 · pp. 4–5

3.3 Training

  • Training uses Adam with analytic time derivatives from Fourier features and autodiff spatial derivatives through the warped Gaussian dictionary; CP-factor normalization fixes gauge symmetry without changing the forward map.Collocation points are periodically resampled, and normalization also improves forward-map conditioning.Source: §3.3 · p. 5

4 Experiments

The experiments evaluate S2-PINN on four controlled random PDEs, compare moment accuracy and calibration with nine baselines, and extend testing to non-manufactured and inverse problems. Results show strong benchmark performance, with projection benefits depending on stochastic structure.

4.1 Benchmark Problems

  • The four controlled benchmarks comprise time-dependent diffusion, Allen–Cahn, and Burgers with a shared manufactured target, plus steady Darcy with exponential random dependence.The comparison uses nine baselines; vanilla PINN has about 28× more parameters than S2-PINN.Source: §4.1 · p. 5

4.2 Metrics

  • Evaluation reports held-out mean and variance relative L2 errors and assesses calibration using empirical coverage at 50%, 80%, 90%, and 95% nominal levels.Moment estimates use 1,024 stochastic samples, and results are averaged over five seeds.Source: §4.2 · pp. 5–6

4.3 Main Results

  • Across diffusion, Allen–Cahn, Burgers, and Darcy, S2-PINN has the lowest observed errors, though its variance advantage is not uniform on Darcy.On the first three polynomial-dependence benchmarks, the projected model beats the best included baseline on mean and variance errors; on Darcy, the no-projection variant gives the best variance.Source: Table 1 · p. 6, Table 2 · p. 6, §4.3 · p. 6
  • S2-PINN remains stable through R=15 random dimensions on the diffusion scaling test and is evaluated at R=50, where its mean and variance relative L2 errors are (1.09±0.14)×10^-2 and (1.24±0.30)×10^-2.The R=50 result is evidence for this diffusion stress test, not a universal high-dimensional guarantee.Source: §4.3 · p. 7
  • S2-PINN closely follows ideal coverage across four manufactured PDEs, while MC-dropout under-covers and the no-projection variant mildly under-covers at 90% and 95%.Near-nominal coverage alone is insufficient when moment accuracy is weaker; a useful stochastic surrogate must match both predictive intervals and solution statistics.Source: §4.3 · pp. 7–8, Figure 2 · p. 7
  • On non-manufactured Poisson, S2-PINN achieves 59× lower mean error and 5× lower variance error than vanilla PINN with 33× fewer parameters; on log-normal Darcy it has the best mean prediction, while deep ensemble has the best variance.Increasing Darcy’s gPC degree from p=2 to p=4 raises SER(p) above 95% and reduces S2-PINN variance error from 15.4% to 4.15%.Source: §4.3 · p. 8, Table 4 · p. 8
  • Projection improves mean accuracy by 2–24% on polynomial-dependence diffusion, Allen–Cahn, and Burgers, but is not uniformly beneficial for exponential-dependence Darcy.The projected residual regularizes toward the polynomial subspace: it can help when the solution is well approximated there and introduce bias otherwise.Source: §4.3 · p. 8, Figure 3 · p. 9

4.4 Stochastic inverse problems

  • Across diffusion and Burgers inverse problems, the approach jointly estimates parameters and stochastic solution fields; for Burgers, the log-mean is recovered within 10^-3 and σ within 5 × 10^-4.Burgers mean solution error is (2.03 ± 0.62) × 10^-3 and variance error is (3.71 ± 0.87) × 10^-3, roughly one order of magnitude above diffusion.Source: §4.4 · p. 10

5 Conclusion

S2-PINN combines stochastic separability with physics-informed learning for forward and inverse UQ, and the paper supports low-rank stochastic separability as an effective modeling principle. Its guarantees rely on stated density conditions and do not provide finite-rank, finite-mode end-to-end solution-error bounds.

5 Conclusion

  • S2-PINN couples Gaussian spatial atoms, Fourier temporal features, and an orthonormal gPC basis through a low-rank CP decomposition for forward and inverse UQ.The method supports closed-form moment evaluation and avoids dense space–time–stochastic grids.Source: §5 · p. 10
  • The method inherits regular-domain bias, and finite rank q and finite gPC size G introduce approximation and stochastic-truncation errors without an end-to-end solution-error guarantee.The paper identifies geometry-aware dictionaries, adaptive rank, and learnable gPC truncation as future directions.Source: §5 · p. 10

A.1 Karhunen–Loève Expansion (KLE)

  • KLE represents a stochastic field using orthogonal spatial eigenfunctions multiplied by uncorrelated random variables, with practical implementations truncating the series to leading terms.The retained terms are chosen to capture the vast majority of field variance.Source: §A.1 · p. 14

A.2 Generalized Polynomial Chaos (gPC)

  • The gPC basis selects orthogonal polynomial families for the input distributions and uses total-degree truncation to avoid a full tensor grid’s exponential growth.For R=10 and p=3, the passage reports 286 modes rather than the full tensor grid’s approximately 10^6 points.Source: §A.2 · pp. 14–15

A.3 Canonical Polyadic (CP) Tensor Decomposition vs. Tensor Train (TT)

  • For the three-way space–time–stochastic representation, CP reduces coupling complexity from dense multiplicative scaling to additive scaling in the factor dimensions and rank.The conclusion gives the resulting complexity as O(q(M + Mt + G)).Source: §A.3 · p. 15

B Theoretical Analysis

  • The theoretical results are asymptotic or estimator-level, assume dense feature families for L2 expressivity, and do not cover finite-budget non-convex optimization.The error decomposition separates CP-rank, gPC-truncation, and Monte Carlo sampling errors.Source: B Theoretical Analysis · p. 16, §B.1 · p. 16

B.2 Expressivity of separable sums

  • Under mild density conditions, the union of finite-rank separable sums is dense in L2 over time, space, and randomness.This follows by identifying the Hilbert tensor product with the product-space L2 space.Source: §B.2 · pp. 16–17

B.3 Projection in the random space

  • The projected residual’s gPC coefficients are expectations against basis functions, and their vanishing is equivalent to the projection being zero.This is the paper’s L2 projection–Galerkin equivalence.Source: §B.3 · pp. 17–18

B.4 Mini-batch estimator and uniform concentration

This section develops mini-batch projection estimates and concentration conditions, relates feature-Gram control and CP factorization to conditioning and efficiency, and presents Gaussian-atom derivative and warp-stability results alongside stochastic PDE benchmarks and reference setups.

B.4 Mini-batch estimator and uniform concentration

  • The projection estimate is unbiased with variance Var(hΨ_g)/B, and uniform error at most ε across G modes holds with high probability when B ≳ σ^2 ε^-2 log(G/δ).The uniform bound assumes the projected features are sub-Gaussian with common proxy σ^2.Source: §B.4 · pp. 19–20
  • The sub-Gaussian condition can fail for log-normal coefficient fields, consistent with reduced variance accuracy on the log-normal Darcy benchmark.The passage says this failure can be detected before training using the subspace energy ratio (8).Source: §B.4 · p. 21

B.5 Feature Gram/Orthogonality conditioning and its concentration

  • If the empirical feature Gram matrix is within δ < 1 of identity, the spatial feature matrix has bounded condition number; bounded features also yield sample-Gram concentration, near identity when population features are orthonormal.The concentration result assumes ∥Φ(x)∥2 ≤ L almost surely.Source: §B.5 · pp. 21, 22, 24

B.6 CP kernel: algebra, efficiency, and gauge

  • The CP kernel uses O((M+M_t+G)q) parameters and O(Bq(M+M_t+G)) batch arithmetic, versus O(MM_tG) parameters and O(BMM_tG) arithmetic for dense coupling.Column-wise normalization fixes the gauge up to signs and permutations and improves Lipschitz conditioning of the forward map with respect to each factor.Source: §B.6 · pp. 24, 26

B.7 Closed-form derivatives for Gaussian atoms and stability under warps

  • Small Jacobian perturbations keep warped Gaussian dictionaries bi-Lipschitz in L2 and preserve near-orthogonality up to a factor controlled by the perturbation.For a residual warp, the chain rule yields derivatives from the warp MLP’s first and second derivatives.Source: §B.7 · p. 26

C Manufactured Solution Definitions

  • The four manufactured benchmarks share domain [-1, 1]^2 × [0, 1], i.i.d. standard-normal variables, and a log-normal random field based on an exponential covariance kernel with correlation length 1.The kernel’s eigenvalue–eigenfunction pairs define the Karhunen–Loève coefficient.Source: C Manufactured Solution Definitions · p. 26

1. Stochastic Diffusion. ∂u

  • The stochastic diffusion benchmark uses a manufactured exact solution, with source obtained by substitution, zero Dirichlet boundary conditions, and matching exact initial data.Source: §1 · p. 27

2. Stochastic Allen–Cahn. ∂u

  • The stochastic Allen–Cahn benchmark uses the diffusion manufactured solution and adds a cubic reaction term, u − u^3.Source: §2 · p. 27

3. Stochastic Burgers. ∂u

  • With viscosity ν = 0.01, stochastic Burgers uses the diffusion manufactured solution; its convective nonlinearity u∇u makes it the most challenging benchmark for standard PINNs.Source: §3 · p. 27

4. Stochastic Darcy (divergence form)

  • The manufactured stochastic Darcy solution couples more tightly to the stochastic field and depends exponentially on Z, requiring higher gPC orders than the first three problems.The passage gives γ = 0.3 and contrasts this exponential dependence with the first three problems’ linear Z-dependence.Source: §4 · p. 27

5. Non-manufactured Stochastic Poisson (forward simulation)

  • The non-manufactured stochastic Poisson problem uses a KL-expanded random source field with σ = 0.5 and R = 5; its reference uses finite differences on a 64 × 64 grid and 10,000 Monte Carlo samples.Source: §4 · p. 27

6. Non-manufactured Stochastic Darcy (forward simulation)

  • The non-manufactured stochastic Darcy problem uses a log-normal permeability field with σ = 1.0 and R = 5; its response is driven by permeability and a constant source, with the same finite-difference Monte Carlo reference procedure.The reference uses 10,000 samples on a 64 × 64 grid.Source: §4 · p. 27

D Details of Baseline Methods

The appendix specifies the compared baseline methods and shared training protocols, including neural, tensor-separable, polynomial-chaos, and classical stochastic approaches.

D Details of Baseline Methods

  • The comparison includes neural and classical baselines, with PC2 added for selected forward benchmarks.The listed comparisons include Vanilla PINN, Vanilla PINN+proj, PI-DeepONet, MC-Dropout PINN, Deep Ensembles, stochastic collocation, SPINN+Z, functional CP-PINN, and Neural Chaos.Source: D Details of Baseline Methods · p. 27
  • Neural baselines generally use Adam, Xavier-normal initialization, resampled batches, and benchmark-specific loss weights and training horizons.The common schedule is 20,000 steps for diffusion, Allen–Cahn, and Burgers, and 30,000 for Darcy; random inputs follow Gaussian or uniform configurations.Source: D Details of Baseline Methods · p. 28
  • Vanilla PINN uses a fully connected MLP on concatenated space, time, and random inputs, without an explicit polynomial basis.For the main R=5 runs, its architecture is (3+R)→256→256→256→256→1 with tanh hidden activations.Source: D Details of Baseline Methods · p. 28
  • Vanilla PINN+proj keeps the Vanilla PINN architecture and adds the gPC Galerkin projection term to isolate the projection objective from tensor separability.Its projection weight is wproj=1.0.Source: D Details of Baseline Methods · p. 28
  • PI-DeepONet combines a stochastic-parameter branch with a spatiotemporal trunk and uses a tuned schedule where the default diverged on Allen–Cahn.Both networks use three hidden layers of width 128 and a 128-dimensional latent representation.Source: D Details of Baseline Methods · p. 28
  • MC-Dropout PINN estimates epistemic uncertainty from 50 inference passes, while Deep Ensembles aggregate five independently trained PINNs.Dropout remains active at inference; ensemble uncertainty is the per-point standard deviation across members.Source: D Details of Baseline Methods · pp. 28–29
  • Stochastic collocation trains an independent deterministic PINN at each tensorized quadrature node, so its run count grows exponentially with random dimension.For R=5, nq=3 requires 243 runs, while the reported tractable baseline uses nq=2 and 32 nodes.Source: D Details of Baseline Methods · p. 29
  • PC2 uses an anisotropic product basis, while SPINN+Z, functional CP-PINN, and Neural Chaos provide distinct separable or learned-basis comparisons.PC2 fixes stochastic degree pZ=3; Neural Chaos is trained on reference realizations and evaluates the physics residual post hoc.Source: D Details of Baseline Methods · p. 29

E Model Architectures, Parameter Counts, and Training Configuration

The section reports architecture and parameter-count comparisons alongside the default hyperparameters used for the four benchmark problems.

E Model Architectures, Parameter Counts, and Training Configuration

  • S2-PINN uses roughly 30× fewer neural parameters than a single Vanilla PINN by coupling compact feature extractors through a CP kernel.Table 8 covers the R=5 stochastic diffusion benchmark and lists architecture details and parameter counts for compared methods and stochastic collocation.Source: E Model Architectures, Parameter… · p. 30, Table 8 · p. 30
  • The benchmark defaults use M=256 spatial atoms, Mt=17 temporal features, and (R,p)=(5,3).The passage gives these settings for the four benchmark problems in Tables 1–2.Source: E Model Architectures, Parameter… · p. 30
  • Training defaults set the PDE, projection, boundary, initial-condition, and orthogonality weights to 1.0, with λfac=0.1 and batch sizes of 32,768.Gradient accumulation κ=1 and resampling period Srs=1 are also specified.Source: E Model Architectures, Parameter… · p. 30

F Ablation Study Details

The ablation study evaluates stochastic diffusion at final time t=1.0 across architecture, spatial atoms, gPC order, and CP rank.

F Ablation Study Details

  • The ablations report errors at final time t=1.0 and cover baseline architecture, spatial atoms, gPC order, and CP rank.The study uses Tables 9–12 for these four ablation dimensions.Source: F Ablation Study Details · p. 31
  • Table 9 covers baseline architecture ablations by hidden size and layer count.Source: Table 9 · p. 31
  • Table 10 covers the number of spatial atoms M.Source: Table 10 · p. 31
  • Table 11 covers gPC order p.Source: Table 11 · p. 31
  • Table 12 covers CP rank q.Source: Table 12 · p. 31

G Scaling Study: Stochastic Dimension and CP Rank

The study sweeps stochastic dimension and CP rank on stochastic diffusion, then extends to an R=50 stress test. Errors remain within reported ranges, while memory rises substantially in the largest run.

G Scaling Study: Stochastic Dimension and CP Rank

H Collocation Density Sensitivity

The collocation-density sweep tests stochastic diffusion at five batch sizes. Mean error improves modestly with larger batches, but calibration is not monotone, motivating a practical default rather than a claim that larger batches are uniformly better.

H Collocation Density Sensitivity

I Computational Cost with Respect to q, M, and R

The cost analysis describes how the CP representation scales with spatial, temporal, and stochastic resolution. It also distinguishes nominal capacity from finite-budget accuracy, which need not improve as rank or basis size grows.

I Computational Cost with Respect to q, M, and R

  • The CP core uses q(M+Mt+G+1) parameters rather than a dense MMtG coupling, and its coupling is cheaper when q≪MMtG/(M+Mt+G).Each added CP component contributes M+Mt+G+1 parameters.Source: I Computational Cost with Respect to… · p. 33
  • For fixed M, Mt, and G, CP parameter count scales as Θ(q) and coupling work as Θ(Bq), although finite-budget accuracy need not improve monotonically with q.The best rank-q approximation error is non-increasing, but training is non-convex and affected by step budget, sampling noise, and regularization.Source: I Computational Cost with Respect to… · p. 33
  • Each additional Gaussian atom adds about q+3 parameters, and spatial feature evaluation and contraction scale linearly with M for fixed remaining sizes.First- and second-order spatial derivatives change the BM term only by a PDE-dependent constant.Source: I Computational Cost with Respect to… · p. 33
  • At fixed polynomial degree p, the CP core grows polynomially in random dimension R rather than exponentially, mitigating but not eliminating the curse of dimensionality.Sparse gPC evaluation reduces the mode-assembly term from RG to pG; the projected residual adds O(BG) work.Source: I Computational Cost with Respect to… · p. 33
  • Increasing R or p reduces truncation error only when newly included coefficients carry non-negligible stochastic energy, while larger spaces increase estimation and optimization difficulty under fixed budgets.The reported comparisons show q=8 competitive for R≤10, q=32 helping at R=15, and p=2 outperforming p=4 in the trained model.Source: I Computational Cost with Respect to… · p. 33

J Stochastic Navier–Stokes Benchmark

The benchmark applies the method to a two-dimensional incompressible Navier–Stokes system in vorticity form with five random inputs. The supplied passages describe the analytic construction and report evaluation of velocity moments and calibration.

J Stochastic Navier–Stokes Benchmark

  • The benchmark uses the 2D incompressible Navier–Stokes equations in vorticity form on [0,2π]^2 over t∈[0,5], with five uniform random variables.The variables control two phase shifts, viscosity, and the amplitudes of two vortex modes.Source: J Stochastic Navier–Stokes Benchmark · p. 34
  • The analytic construction satisfies incompressibility exactly, while the two vortex modes produce a non-zero nonlinear interaction.The linear time-derivative and diffusion terms cancel in the construction.Source: J Stochastic Navier–Stokes Benchmark · p. 34
  • The reported evaluations cover relative L2 errors for velocity mean and variance, plus velocity calibration coverage and calibration error.Cov90 and Cov95 are judged against 0.900 and 0.950, respectively, and smaller calibration error is better.Source: Table 15 · p. 34, Table 16 · p. 34

K Additional Baseline Results

This section presents additional baseline calibration and resource measurements, visualizations of stochastic-field predictions and errors, and checklist disclosures about human subjects and LLM use.

K Additional Baseline Results

  • Tables 17 and 18 report additional-baseline calibration and training-time and peak-memory measurements, respectively.Table 17 covers PI-DeepONet calibration across all metrics and calibration error for SPINN+Z, functional CP-PINN, and Neural Chaos; numerical values are not supplied here.Source: K Additional Baseline Results · p. 34, Table 17 · p. 40, Table 18 · p. 40

L Visualizations of Stochastic Fields

NeurIPS Paper Checklist

  • The checklist reports that experiments average five random seeds with mean and standard deviation, use NVIDIA T4 GPUs, and provide reproducibility details in Algorithm 1 and Appendix E.It also says stability and consistency proofs appear in Appendix B, limitations and societal impacts are discussed in Section 5, and an anonymous code link is provided.Source: NeurIPS Paper Checklist · pp. 42–44
  • The research did not involve human subjects, so the checklist marks IRB review as not applicable.Source: NeurIPS Paper Checklist · p. 46
  • The research did not use LLMs as an important, original, or non-standard component of its core methods.Source: NeurIPS Paper Checklist · p. 47

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