Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Data

TL;DR

This work addresses the challenge of solving forward and inverse PDEs from noisy data with quantified uncertainty. It introduces BPIELM, which merges physics-informed constraints with a Bayesian prior over output weights, yielding posterior distributions that quantify prediction uncertainty while solving linear PDEs efficiently. Compared to PIELM and PINN, BPIELM offers improved predictive accuracy under noise, robust performance across forward and inverse tasks (including Poisson, advection, diffusion, and Helmholtz equations), and drastically reduced computational cost. The approach is particularly advantageous when uncertainty quantification is critical and data are corrupted by noise, enabling fast, reliable PDE inference with unified treatment of unknown parameters in inverse problems.

Abstract

Physics-informed extreme learning machine (PIELM) has recently received significant attention as a rapid version of physics-informed neural network (PINN) for solving partial differential equations (PDEs). The key characteristic is to fix the input layer weights with random values and use Moore-Penrose generalized inverse for the output layer weights. The framework is effective, but it easily suffers from overfitting noisy data and lacks uncertainty quantification for the solution under noise scenarios.To this end, we develop the Bayesian physics-informed extreme learning machine (BPIELM) to solve both forward and inverse linear PDE problems with noisy data in a unified framework. In our framework, a prior probability distribution is introduced in the output layer for extreme learning machine with physic laws and the Bayesian method is used to estimate the posterior of parameters. Besides, for inverse PDE problems, problem parameters considered as new output layer weights are unified in a framework with forward PDE problems. Finally, we demonstrate BPIELM considering both forward problems, including Poisson, advection, and diffusion equations, as well as inverse problems, where unknown problem parameters are estimated. The results show that, compared with PIELM, BPIELM quantifies uncertainty arising from noisy data and provides more accurate predictions. In addition, BPIELM is considerably cheaper than PINN in terms of the computational cost.

Figures (17)

• Figure 1: Conceptual flow of the BPIELM method with noisy data. $p(\boldsymbol{\omega} \mid \eta)$ is the prior for the output layer weights as well as problem parameters, $p\left(\mathbf{Y} \mid \boldsymbol{\omega}, \sigma^{2}\right)$ represents the likelihood of measurements, and $p\left(\boldsymbol{\omega} \mid \mathbf{Y},\eta, \sigma^{2}\right)$ represents the posterior.
• Figure 2: Poisson equation: The first row represents the exact solution, the PIELM solution and its absolute error, respectively. The second row represents the standard deviations for u using BPIELM, the BPIELM solution and its absolute error, respectively. The noise scale on the boundary is $\epsilon_{b} \sim \mathcal{N}\left(0,0.01^{2}\right)$, the number of neurons is $N=100$, and the number of training points is $N_{f}=400$ and $N_{b}=38$. Black circles represent the positions of sensors.
• Figure 3: Poisson equation: The columns (from left to right) represent comparisons between BPIELM and PIELM at $y=-0.3$, $y=0$ and $y = 0.4$, where $N=100$, $N_{f}=400$ and $N_{b}=41$ are used. The row (from top to down) represents two noise scales on the boundary, $\epsilon_{b} \sim \mathcal{N}\left(0,0.01^{2}\right)$ and $\epsilon_{b} \sim \mathcal{N}\left(0,0.1^{2}\right)$.
• Figure 4: Poisson equation: The first column represents the Max-AE and the MAE under different number of sensors on the boundary, where $\epsilon_{b} \sim \mathcal{N}\left(0,0.01^{2}\right)$, $N=100$ and $N_{f}=200$ are used. The second and third columns represent the predictions at $N_{b}=19$ and its absolute error. The rows (from top to down) represent the results of PIELM and BPIELM. Black circles represent the positions of sensors.
• Figure 5: Poisson equation: The columns (from left to right) represent the MAE comparisons between BPIELM and PIELM at $N_{bc}=21$, $N_{bc}=42$ and $N_{bc}=63$, where $\epsilon_{b} \sim \mathcal{N}\left(0,0.01^{2}\right)$ and $N_{f}=400$ are used.