Research map / Probabilistic and scientific modeling / Differential equations and PDEs

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

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Physics-informed neural network training · 0 citations (OpenAlex)

Read the paperCheck your idea against 108,967 papers

AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.

Proposes S²-PINN, a physics-informed neural solver for random PDEs that combines a learnable Gaussian spatial dictionary, Fourier temporal features, and gPC stochastic modes via a low-rank CP core, trained with a hybrid strong-form and gPC-projected residual loss.

Problem

Uncertainty quantification for random PDEs: classical spectral solvers suffer the curse of dimensionality and existing neural solvers ignore stochastic structure needed for accurate moments and calibration.

Approach

Represents the random PDE solution as a rank-q CP-coupled product of learnable spatial Gaussian atoms, Fourier temporal features, and orthonormal gPC stochastic modes, and adds a gPC-projected residual term that corresponds to a stochastic Galerkin constraint.

Key results

• On manufactured benchmarks S²-PINN (proj) reports mean relative L2 errors of about 3.4e-3 to 4.4e-3 on diffusion, Allen-Cahn and Burgers, versus 1.41e-2 to 1.81e-2 for vanilla PINN.
• On the non-manufactured Poisson problem S²-PINN achieves 59x lower mean error than the vanilla PINN with 33x fewer parameters (6,114 vs 199,937).
• Calibration: S²-PINN (proj) gives near-nominal coverage (e.g. Cov@90% ≈ 0.897–0.901) with calibration error ≈ 0.005–0.006 on the four manufactured PDEs, while MC-Dropout under-covers.

Surprising findings

• On log-normal Darcy the no-projection variant gives better mean and variance than the projected variant, showing the gPC projection can introduce bias outside the polynomial regime.
• Adding the gPC projection to a vanilla MLP PINN degrades several benchmarks, indicating the benefit depends on the separable architecture.

Limitations

• Theory gives only asymptotic expressivity and estimator-level results, with no finite-rank, finite-gPC-order end-to-end solution error bound.
• Restricted to regular domains inherited from separable coordinate architectures.
• Projection is not uniformly beneficial; log-normal Darcy variance error remains larger than deep ensemble before increasing gPC degree.

Experimental setup

Datasets: manufactured random PDE benchmarks (diffusion, Allen-Cahn, Burgers, Darcy) stochastic Poisson and log-normal Darcy finite-difference Monte Carlo references stochastic Navier-Stokes

Benchmarks: diffusion Allen-Cahn Burgers Darcy stochastic Poisson log-normal Darcy stochastic Navier-Stokes diffusion scaling R=50 stochastic inverse diffusion and Burgers

Metrics: mean relative L2 error variance relative L2 error empirical coverage sharpness calibration error

Baselines: Vanilla PINN Vanilla PINN + proj PI-DeepONet MC-Dropout PINN Deep Ensemble (K=5) Stochastic collocation (32 node-PINNs) SPINN+Z Functional CP-PINN Neural Chaos PC^2 (UQpy LAR/OLS)

Builds on: SPINN PINN generalized polynomial chaos deep ensembles PI-DeepONet

Closest papers by meaning

• Efficient Error Certification for Physics-Informed Neural Networks (ICML 2024 · similarity 0.89)
• Sub-Sequential Physics-Informed Learning with State Space Model (ICML 2025 · similarity 0.89)
• Parameterized Physics-informed Neural Networks for Parameterized PDEs (ICML 2024 · similarity 0.88)
• SPDEBench: An Extensive Benchmark for Learning Stochastic PDEs (NeurIPS 2026 · similarity 0.88)
• The Challenges of the Nonlinear Regime for Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.87)
• RoPINN: Region Optimized Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.87)