Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Solving forward and inverse PDE problems on unknown manifolds via physics-informed neural operators

Anran Jiao, Qile Yan, Jhn Harlim, Lu Lu

TL;DR

This work introduces DeepONet and physics-informed DeepONet (PI-DeepONet) to learn PDE solution operators on unknown manifolds identified by point clouds, enabling both forward and inverse problem solving. The authors compare three operator-approximation schemes (Diffusion Maps, Radial Basis Functions, and Generalized Moving Least Squares) for discretizing tangential differential operators on manifolds and quantify the induced error in the PI-DeepONet loss. Numerical experiments on torus and semi-torus geometries demonstrate that PI-DeepONet substantially improves accuracy, especially when observation data are limited, and remains robust across diffusion coefficients and boundary conditions. They further embed PI-DeepONet into a Bayesian MCMC framework to infer diffusion coefficients from noisy PDE observations, achieving comparable accuracy to local PDE solves but with orders-of-magnitude faster per-iteration performance. Overall, the framework offers a scalable, data-driven approach for forward and inverse PDE problems on unknown manifolds with substantial computational gains for Bayesian inference.

Abstract

In this paper, we evaluate the effectiveness of deep operator networks (DeepONets) in solving both forward and inverse problems of partial differential equations (PDEs) on unknown manifolds. By unknown manifolds, we identify the manifold by a set of randomly sampled data point clouds that are assumed to lie on or close to the manifold. When the loss function incorporates the physics, resulting in the so-called physics-informed DeepONets (PI-DeepONets), we approximate the differentiation terms in the PDE by an appropriate operator approximation scheme. For the second-order elliptic PDE with a nontrivial diffusion coefficient, we approximate the differentiation term with one of these methods: the Diffusion Maps (DM), the Radial Basis Functions (RBF), and the Generalized Moving Least Squares (GMLS) methods. For the GMLS approximation, which is more flexible for problems with boundary conditions, we derive the theoretical error bound induced by the approximate differentiation. Numerically, we found that DeepONet is accurate for various types of diffusion coefficients, including linear, exponential, piecewise linear, and quadratic functions, for linear and semi-linear PDEs with/without boundaries. When the number of observations is small, PI-DeepONet trained with sufficiently large samples of PDE constraints produces more accurate approximations than DeepONet. For the inverse problem, we incorporate PI-DeepONet in a Bayesian Markov Chain Monte Carlo (MCMC) framework to estimate the diffusion coefficient from noisy solutions of the PDEs measured at a finite number of point cloud data. Numerically, we found that PI-DeepONet provides accurate approximations comparable to those obtained by a more expensive method that directly solves the PDE on the proposed diffusion coefficient in each MCMC iteration.

Solving forward and inverse PDE problems on unknown manifolds via physics-informed neural operators

TL;DR

This work introduces DeepONet and physics-informed DeepONet (PI-DeepONet) to learn PDE solution operators on unknown manifolds identified by point clouds, enabling both forward and inverse problem solving. The authors compare three operator-approximation schemes (Diffusion Maps, Radial Basis Functions, and Generalized Moving Least Squares) for discretizing tangential differential operators on manifolds and quantify the induced error in the PI-DeepONet loss. Numerical experiments on torus and semi-torus geometries demonstrate that PI-DeepONet substantially improves accuracy, especially when observation data are limited, and remains robust across diffusion coefficients and boundary conditions. They further embed PI-DeepONet into a Bayesian MCMC framework to infer diffusion coefficients from noisy PDE observations, achieving comparable accuracy to local PDE solves but with orders-of-magnitude faster per-iteration performance. Overall, the framework offers a scalable, data-driven approach for forward and inverse PDE problems on unknown manifolds with substantial computational gains for Bayesian inference.

Abstract

In this paper, we evaluate the effectiveness of deep operator networks (DeepONets) in solving both forward and inverse problems of partial differential equations (PDEs) on unknown manifolds. By unknown manifolds, we identify the manifold by a set of randomly sampled data point clouds that are assumed to lie on or close to the manifold. When the loss function incorporates the physics, resulting in the so-called physics-informed DeepONets (PI-DeepONets), we approximate the differentiation terms in the PDE by an appropriate operator approximation scheme. For the second-order elliptic PDE with a nontrivial diffusion coefficient, we approximate the differentiation term with one of these methods: the Diffusion Maps (DM), the Radial Basis Functions (RBF), and the Generalized Moving Least Squares (GMLS) methods. For the GMLS approximation, which is more flexible for problems with boundary conditions, we derive the theoretical error bound induced by the approximate differentiation. Numerically, we found that DeepONet is accurate for various types of diffusion coefficients, including linear, exponential, piecewise linear, and quadratic functions, for linear and semi-linear PDEs with/without boundaries. When the number of observations is small, PI-DeepONet trained with sufficiently large samples of PDE constraints produces more accurate approximations than DeepONet. For the inverse problem, we incorporate PI-DeepONet in a Bayesian Markov Chain Monte Carlo (MCMC) framework to estimate the diffusion coefficient from noisy solutions of the PDEs measured at a finite number of point cloud data. Numerically, we found that PI-DeepONet provides accurate approximations comparable to those obtained by a more expensive method that directly solves the PDE on the proposed diffusion coefficient in each MCMC iteration.
Paper Structure (22 sections, 2 theorems, 50 equations, 7 figures, 5 tables)

This paper contains 22 sections, 2 theorems, 50 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. DeepONet
  4. PI-DeepONet
  5. Error induced by the approximate loss function
  6. Numerical experiments
  7. Second-order linear elliptic PDE on torus
  8. Results from DeepONet
  9. Results from PI-DeepONet
  10. Second-order linear elliptic PDE on semi-torus with Dirichlet boundary conditions
  11. Nonlinear PDE on torus
  12. Application to solving Bayesian inverse problems
  13. Bayesian approach to the inverse problem
  14. Prior specification and discretization
  15. Approximation of the forward map
  16. ...and 7 more sections

Key Result

Lemma 3.1

Let $X^{(k)} = \{\mathbf{x}_1^{(k)},\ldots,\mathbf{x}_N^{(k)}\}$ be a set of uniformly i.i.d. samples of $M$. Let $\kappa^{(k)} \in C^1(M)$ and the estimator, $\mathcal{G}_\theta(\kappa^{(k)}(X^{(k)}))\in C^{p+1}(M^*)$ function, where $M^* = \bigcup_{\mathbf{x}\in M} B(\mathbf{x},C_2N^{-1/d})$ is a

Figures (7)

  • Figure 1: The architecture of DeepONets. (A) DeepONet. (B) PI-DeepONet.
  • Figure 2: DeepONet predictions for the linear $\kappa$ in Section \ref{['sec:torus_deeponet']}. (A) Examples of linear $\kappa$, exponential $\kappa$, piecewise $\kappa$, and higher order $\kappa$.(B) 2D and 3D visualizations of one linear $\kappa$. (C) 2D and 3D visualizations of the solution $u$ generated by Diffusion Maps. (D) Predicted solutions and the absolute error with $N_{OBS}=1000, 500$, and $100$.
  • Figure 3: An example of prediction of DeepONet and PI-DeepONet in Section \ref{['sec:torus_pideeponet']}. (A) The 2D and 3D visualizations of $\kappa$. (B) The 2D visualization of solutions generated by Diffusion Maps and RBF. (C) The prediction and absolute error of the methods with $N_{OBS}=10$ and 25 for observation loss using Diffusion Maps approach. (D) The prediction and absolute error of the methods with $N_{OBS}=10$ and 25 for observation loss using RBF approach.
  • Figure 4: Predictions of DeepONet and PI-DeepONet with GMLS approach in Section \ref{['sec:linear_semitorus']}. (A) The 2D and 3D visualizations of $\kappa$. (B) The 2D and 3D visualizations of solution generated by GMLS. (C) The prediction and absolute error of the methods with $N_{OBS}=2$ and $10$ for observation loss.
  • Figure 5: Predictions of DeepONet and PI-DeepONet with Diffusion Maps approach in Section \ref{['sec:nonlinear_torus']}. (A) The 2D and the 3D visualizations of $\kappa$. (B) The 2D and the 3D visualizations of the solution generated by Diffusion Maps. (C) The prediction and absolute error of the methods with $N_{OBS}=2$ and $10$.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Lemma 3.1
  • Proposition 3.2
  • proof