Accurate adaptive deep learning method for solving elliptic problems

TL;DR

This paper targets accurate and efficient solving of elliptic PDEs with deep learning, addressing interface and convection-dominated scenarios. It combines Levenberg–Marquardt optimization with a failure-probability–driven adaptive sampling framework, and introduces a piece-wise uniform approximation (PUA) of the optimal proposal distribution together with a kernel-based sampling mechanism. An extension–projection approach handles elliptic interfaces, enabling a unified treatment across problem types. Across 2D/5D Poisson problems, elliptic interface problems, and convection-dominated cases, the method achieves relative $L^2$ errors improved by 2–4 orders of magnitude and outperforms SAIS-based adaptive sampling, demonstrating substantial gains in accuracy and efficiency for physics-informed DL solvers.

Abstract

Deep learning method is of great importance in solving partial differential equations. In this paper, inspired by the failure-informed idea proposed by Gao et.al. (SIAM Journal on Scientific Computing 45(4)(2023)) and as an improvement, a new accurate adaptive deep learning method is proposed for solving elliptic problems, including the interface problems and the convection-dominated problems. Based on the failure probability framework, the piece-wise uniform distribution is used to approximate the optimal proposal distribution and an kernel-based method is proposed for efficient sampling. Together with the improved Levenberg-Marquardt optimization method, the proposed adaptive deep learning method shows great potential in improving solution accuracy. Numerical tests on the elliptic problems without interface conditions, on the elliptic interface problem, and on the convection-dominated problems demonstrate the effectiveness of the proposed method, as it reduces the relative errors by a factor varying from $10^2$ to $10^4$ for different cases.

Figures (10)

• Figure 1: Schematic diagram of PUA. The gray area denotes the failure domain, and the black points within this domain indicate the locations where PINNs fail. The red stars represent new sampling points obtained by PUA from the optimal proposal distribution generated by the transition kernel $K(\cdot)$.
• Figure 2: The relative $L^2$ errors for example \ref{['example1']} obtained by two different adaptive sampling methods.
• Figure 3: Contour plots of residuals of the loss function and distributions of new points obtained by PUA and SAIS for four iterations.
• Figure 4: Left: The solution profile obtained by PUA. Right: Profile of the pointwise errors. Here the relative $L^2$ error is $8.41\times 10^{-6}$.
• Figure 5: Contour plots of residuals of the loss function and distributions of new points obtained by PUA for four iterations.