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Physics-informed neural networks for operator equations with stochastic data

Paul Escapil-Inchauspé, Gonzalo A. Ruz

TL;DR

This work extends physics-informed neural networks to operator equations with stochastic data by formulating and solving tensor operator equations for statistical moments. It introduces two TPINN architectures—vanilla (V-TPINN) and multi-output (MO-TPINN)—and provides a theoretical generalization bound for V-TPINNs, linking training error and quadrature error to the overall accuracy. Through extensive numerical experiments on Poisson, Schrödinger, Helmholtz, and Heat problems, the paper demonstrates that TPINNs can accurately compute higher-order moments with modest collocation counts, while revealing trade-offs in computational cost and derivative requirements as $k$ grows. The results suggest that MO-TPINNs offer practical advantages in certain settings, and the framework establishes a foundation for future convergence analyses and UQ-enabled PDE workflows using tensor operator equations.

Abstract

We consider the computation of statistical moments to operator equations with stochastic data. We remark that application of PINNs -- referred to as TPINNs -- allows to solve the induced tensor operator equations under minimal changes of existing PINNs code, and enabling handling of non-linear and time-dependent operators. We propose two types of architectures, referred to as vanilla and multi-output TPINNs, and investigate their benefits and limitations. Exhaustive numerical experiments are performed; demonstrating applicability and performance; raising a variety of new promising research avenues.

Physics-informed neural networks for operator equations with stochastic data

TL;DR

This work extends physics-informed neural networks to operator equations with stochastic data by formulating and solving tensor operator equations for statistical moments. It introduces two TPINN architectures—vanilla (V-TPINN) and multi-output (MO-TPINN)—and provides a theoretical generalization bound for V-TPINNs, linking training error and quadrature error to the overall accuracy. Through extensive numerical experiments on Poisson, Schrödinger, Helmholtz, and Heat problems, the paper demonstrates that TPINNs can accurately compute higher-order moments with modest collocation counts, while revealing trade-offs in computational cost and derivative requirements as grows. The results suggest that MO-TPINNs offer practical advantages in certain settings, and the framework establishes a foundation for future convergence analyses and UQ-enabled PDE workflows using tensor operator equations.

Abstract

We consider the computation of statistical moments to operator equations with stochastic data. We remark that application of PINNs -- referred to as TPINNs -- allows to solve the induced tensor operator equations under minimal changes of existing PINNs code, and enabling handling of non-linear and time-dependent operators. We propose two types of architectures, referred to as vanilla and multi-output TPINNs, and investigate their benefits and limitations. Exhaustive numerical experiments are performed; demonstrating applicability and performance; raising a variety of new promising research avenues.
Paper Structure (20 sections, 1 theorem, 67 equations, 14 figures, 8 tables)

This paper contains 20 sections, 1 theorem, 67 equations, 14 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Operator equations with stochastic data
  3. General Notation
  4. Abstract problem
  5. TPINNs
  6. V-TPINNs
  7. MO-TPINNs
  8. Convergence analysis for V-TPINNs
  9. Implementation
  10. Numerical experiments
  11. Methodology
  12. Poisson 1D
  13. Stationary 1D Schrödinger
  14. Helmholtz equation in two dimensions
  15. Separable right-hand side
  16. ...and 5 more sections

Key Result

Theorem 4.1

\newlabelthm:boundGeneralization0 Under the present setting, the generalization error in eq:genErrorDef is bounded as: with $\gamma_\mathsf{L}$ the stability constant in eq:continuousdep1 and $c_\textup{quad}$ in eq:boundQuadrature.

Figures (14)

  • Figure 1: Schematic representation of a V-TPINN. The neural networks have $L-1=2$ hidden layers.
  • Figure 1: Illustration of the total error for TPINNs. In \ref{['thm:boundGeneralization']} we provide a bound for the generalization error for V-TPINNs.
  • Figure 1: Flowchart of DeepXDE in lu2021deepxde showing that TPINNs can be added to PINNs code with minor changes. The additions to the source code are in red. Boxes in green are adapted to the tensor setting in the main code via proper definition of the loss function and hard constraints.
  • Figure 1: Poisson 1D. $\text{diag}(\Sigma^{\textup{k}}_\theta)$ for TPINNs (up, purple) and MO-TPINNs (bottom, pink) for ${\textup{k}}=1,\cdots,4$. The exact solution is plotted with dashed green lines.
  • Figure 2: Schematic representation of a MO-TPINN. The neural networks has $L-1=2$ hidden layers.
  • ...and 9 more figures

Theorems & Definitions (3)

  • Remark 3.1: Hard BCs
  • Theorem 4.1: Generalization error for V-TPINNs
  • Proof 1