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Monte Carlo-Type Neural Operator for Differential Equations

Salah Eddine Choutri, Prajwal Chauhan, Othmane Mazhar, Saif Eddin Jabari

TL;DR

This paper proposes MCNO, a Monte Carlo-type Neural Operator that learns kernel operators for PDE solution mappings without relying on translation invariance or global spectral bases. It uses a one-time Monte Carlo sampling of grid points to approximate the kernel integral, coupled with feature-channel mixing and a grid-interpolating step to support arbitrary discretizations, achieving linear computational complexity. Empirical results on Burgers' and KdV equations show MCNO is competitive with, and often superior to, Fourier- and wavelet-based neural operators, while providing theoretical bounds on MC estimation bias and variance that extend to higher dimensions. The work suggests Monte Carlo-based neural operators as scalable, theoretically grounded alternatives for nonlinear operator learning and points to future extensions to higher-dimensional and unstructured domains.

Abstract

The Monte Carlo-type Neural Operator (MCNO) introduces a framework for learning solution operators of one-dimensional partial differential equations (PDEs) by directly learning the kernel function and approximating the associated integral operator using a Monte Carlo-type approach. Unlike Fourier Neural Operators (FNOs), which rely on spectral representations and assume translation-invariant kernels, MCNO makes no such assumptions. The kernel is represented as a learnable tensor over sampled input-output pairs, and sampling is performed once, uniformly at random from a discretized grid. This design enables generalization across multiple grid resolutions without relying on fixed global basis functions or repeated sampling during training, while an interpolation step maps between arbitrary input and output grids to further enhance flexibility. Experiments on standard 1D PDE benchmarks show that MCNO achieves competitive accuracy with efficient computational cost. We also provide a theoretical analysis proving that the Monte Carlo estimator yields a bounded bias and variance under mild regularity assumptions. This result holds in any spatial dimension, suggesting that MCNO may extend naturally beyond one-dimensional problems. More broadly, this work explores how Monte Carlo-type integration can be incorporated into neural operator frameworks for continuous-domain PDEs, providing a theoretically supported alternative to spectral methods (such as FNO) and to graph-based Monte Carlo approaches (such as the Graph Kernel Neural Operator, GNO).

Monte Carlo-Type Neural Operator for Differential Equations

TL;DR

This paper proposes MCNO, a Monte Carlo-type Neural Operator that learns kernel operators for PDE solution mappings without relying on translation invariance or global spectral bases. It uses a one-time Monte Carlo sampling of grid points to approximate the kernel integral, coupled with feature-channel mixing and a grid-interpolating step to support arbitrary discretizations, achieving linear computational complexity. Empirical results on Burgers' and KdV equations show MCNO is competitive with, and often superior to, Fourier- and wavelet-based neural operators, while providing theoretical bounds on MC estimation bias and variance that extend to higher dimensions. The work suggests Monte Carlo-based neural operators as scalable, theoretically grounded alternatives for nonlinear operator learning and points to future extensions to higher-dimensional and unstructured domains.

Abstract

The Monte Carlo-type Neural Operator (MCNO) introduces a framework for learning solution operators of one-dimensional partial differential equations (PDEs) by directly learning the kernel function and approximating the associated integral operator using a Monte Carlo-type approach. Unlike Fourier Neural Operators (FNOs), which rely on spectral representations and assume translation-invariant kernels, MCNO makes no such assumptions. The kernel is represented as a learnable tensor over sampled input-output pairs, and sampling is performed once, uniformly at random from a discretized grid. This design enables generalization across multiple grid resolutions without relying on fixed global basis functions or repeated sampling during training, while an interpolation step maps between arbitrary input and output grids to further enhance flexibility. Experiments on standard 1D PDE benchmarks show that MCNO achieves competitive accuracy with efficient computational cost. We also provide a theoretical analysis proving that the Monte Carlo estimator yields a bounded bias and variance under mild regularity assumptions. This result holds in any spatial dimension, suggesting that MCNO may extend naturally beyond one-dimensional problems. More broadly, this work explores how Monte Carlo-type integration can be incorporated into neural operator frameworks for continuous-domain PDEs, providing a theoretically supported alternative to spectral methods (such as FNO) and to graph-based Monte Carlo approaches (such as the Graph Kernel Neural Operator, GNO).

Paper Structure

This paper contains 21 sections, 1 theorem, 26 equations, 9 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Learning Neural Operators
  3. Operator Approximation
  4. Iterative Update Framework.
  5. Kernel Integral Operator.
  6. MC-type Neural Operator Architecture
  7. Monte Carlo Approximation for Integral Operators
  8. Efficient kernel interactions and feature mixing
  9. Interpolation for grid reconstruction
  10. Linear Complexity
  11. Bounding the Monte Carlo estimation error
  12. Numerical experiments
  13. Burgers' Equation
  14. Dataset and Problem Setup
  15. Results and Comparisons
  16. ...and 6 more sections

Key Result

Theorem 3.2

The Monte Carlo estimator $\hat{\mathcal{K}}_{N, \phi} v_t(x)$ approximates the integral operator $\left( \mathcal{K}_\phi v_t \right)(x)$ by averaging over $N$ points $\{y_{s_i}\}_{i=1}^N$ sampled uniformly at random from the grid and under the assumptions assum:RegultarityOfKv we get with probabil

Figures (9)

  • Figure 1: MCNO's architecture
  • Figure 2: $L_2$ test loss vs Number of samples for Burgers
  • Figure 3: Time per epoch vs Number of samples for Burgers
  • Figure 4: $L_2$ test loss vs Number of samples for KdV
  • Figure 5: Time per epoch vs Number of samples for KdV
  • ...and 4 more figures

Theorems & Definitions (2)

  • Theorem 3.2
  • proof