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Neural Measures for learning distributions of Random PDEs

Georgios Arampatzis, Stylianos Katsarakis, Charalambos Makridakis

TL;DR

This work develops a variational, neural-measure framework to learn the distribution of solutions to random PDEs by representing solution laws in the Bochner space $L^2(\\gamma;\\mathcal{H})$ and learning pushforward measures via neural encodings. It introduces discrete neural measure spaces in three architectures—Fully Network Based, PCE-NN, and Galerkin-NN—to approximate $\\mathcal{P}_2^\\gamma(\\mathcal{H})$, and optimises Wasserstein-based losses that couple transformed solution and source laws through operators $A_\\xi$ and $B_\\xi$. The paper provides universal approximation results for neural measures in Banach/Hilbert settings and demonstrates the method on bistable ODE, diffusion, and reaction-diffusion RPDEs, showing accurate mean and higher-order statistics while highlighting challenges with sharply peaked distributions. Overall, the framework advances uncertainty quantification for random PDEs by coupling PINN-inspired physics with neural generative measures and optimal transport-based learning, offering scalable tools for uncertainty-aware predictions in high-dimensional spaces.

Abstract

The integration of Scientific Machine Learning (SciML) techniques with uncertainty quantification (UQ) represents a rapidly evolving frontier in computational science. This work advances Physics-Informed Neural Networks (PINNs) by incorporating probabilistic frameworks to effectively model uncertainty in complex systems. Our approach enhances the representation of uncertainty in forward problems by combining generative modeling techniques with PINNs. This integration enables in a systematic fashion uncertainty control while maintaining the predictive accuracy of the model. We demonstrate the utility of this method through applications to random differential equations and random partial differential equations (PDEs).

Neural Measures for learning distributions of Random PDEs

TL;DR

This work develops a variational, neural-measure framework to learn the distribution of solutions to random PDEs by representing solution laws in the Bochner space and learning pushforward measures via neural encodings. It introduces discrete neural measure spaces in three architectures—Fully Network Based, PCE-NN, and Galerkin-NN—to approximate , and optimises Wasserstein-based losses that couple transformed solution and source laws through operators and . The paper provides universal approximation results for neural measures in Banach/Hilbert settings and demonstrates the method on bistable ODE, diffusion, and reaction-diffusion RPDEs, showing accurate mean and higher-order statistics while highlighting challenges with sharply peaked distributions. Overall, the framework advances uncertainty quantification for random PDEs by coupling PINN-inspired physics with neural generative measures and optimal transport-based learning, offering scalable tools for uncertainty-aware predictions in high-dimensional spaces.

Abstract

The integration of Scientific Machine Learning (SciML) techniques with uncertainty quantification (UQ) represents a rapidly evolving frontier in computational science. This work advances Physics-Informed Neural Networks (PINNs) by incorporating probabilistic frameworks to effectively model uncertainty in complex systems. Our approach enhances the representation of uncertainty in forward problems by combining generative modeling techniques with PINNs. This integration enables in a systematic fashion uncertainty control while maintaining the predictive accuracy of the model. We demonstrate the utility of this method through applications to random differential equations and random partial differential equations (PDEs).

Paper Structure

This paper contains 24 sections, 2 theorems, 107 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Approach
  3. The problem and its probabilistic framework
  4. Problem and Assumptions
  5. Probabilistic framework
  6. Transformed Law due to Parameterised Operators
  7. Variational Approximation of the Target Law
  8. Discrete Neural Measure Spaces
  9. Fully Network Based Neural Measure Spaces
  10. PCE-NN Based Neural Measure Spaces
  11. Galerkin-NN Based Neural Measure Spaces
  12. Distances and their evaluation
  13. Loss based on Wasserstein distance
  14. Other metrics and divergences
  15. Divergences and Metrics: Definitions and Relationships
  16. ...and 9 more sections

Key Result

Theorem A.2

Suppose we are given parameters $k,\,\beta$, a Borel subset $U\subset\mathbb{R}^m$ and a Banach space $X$ consisting of functions with domain $U$ such that Ass:UAT:Banach holds. Let $\rho\in \overline{C_b^k(\mathbb{R})}^\beta$ be a non-polynomial activation function, and consider the following class and let $\mathcal{N}_{\rho,\mathbb{R}^m,d}\coloneqq \bigcup_{K\in\mathbb{N}}\mathcal{N}_{\rho,\math

Figures (6)

  • Figure 1: Histogram comparison at time $t=2$ of the solution of the bistable ODE \ref{['eq:bistable_ode']} using an implicit numerical solver and the PINN architecture for different parameter values.
  • Figure 1: Training and testing loss over time during the training of the PINN architecture for the bistable ODE \ref{['eq:bistable_ode']}.
  • Figure 2: Average and one standard deviation of the reference and PINN-based solutions (a), at time $t=0.5$ and $t=1$. Average values (b) and histograms of the solution (c) of the diffusion equation \ref{['eq:diffusion']} at time $t=0.5$ using the PINN and the PINN-PCE architectures. Notice that PINN produces marginally better results than the PINN-PCE architecture.
  • Figure 2: Training and testing loss function over time of the training of the PINN and the PINN-PCE architecture for the diffusion equation \ref{['eq:diffusion']}.
  • Figure 3: Average values and histograms of the solution of the reaction-diffusion equation \ref{['eq:reaction_diffusion']} at time $t = 2$ using the PINN architecture. The statistics produced by the PINN architecture are in good qualitative agreement with the reference solution.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Theorem A.2: Restating Theorem 2.7 in neufeld2025universalapproximationresultsneural, UAT for approximating Banach space $X$
  • Theorem A.4: Universal Approximation via PCE-NN in Neural Measure Spaces
  • Remark A.5
  • Proof 1