DAS-PINNs for high-dimensional partial differential equations: extending deep adaptive sampling to spacetime domains

2026-06-04

AI Summary

This work addresses the challenge of efficiently solving high-dimensional spatiotemporal partial differential equations (PDEs), whose solutions often exhibit localized and dynamically evolving features that render conventional physics-informed neural networks (PINNs) inefficient under uniform sampling. To overcome this limitation, the authors propose an end-to-end adaptive sampling framework that treats space and time as a unified domain. By integrating PINNs with normalizing flow models, the method learns the distribution of PDE residuals to automatically identify and concentrate sampling in high-residual regions—without requiring explicit time-stepping or moving meshes. The approach is validated across multiple benchmark problems, ranging from two-dimensional sharp moving fronts to localized structures in eight-dimensional space, demonstrating substantial improvements in both accuracy and computational efficiency.

Abstract

Time-dependent high-dimensional partial differential equations (PDEs) with spatially localised and dynamically evolving solutions pose a fundamental challenge for physics-informed neural networks (PINNs), as uniform collocation sampling becomes increasingly ineffective in high-dimensional spatiotemporal domains. In this work, a deep adaptive sampling framework for PINNs is extended to the time-dependent setting by treating space and time as a unified domain without any explicit time marching. A normalising flow neural network model effectively learns the distribution induced by the PDE residual and generates new collocation points concentrated in regions where the solution is most difficult to learn. Unlike conventional adaptive strategies that require explicit time stepping or moving meshes, high-residual regions are automatically identified and tracked across both space and time, driven purely by the PDE residual distribution. The effectiveness of the proposed strategy is assessed on a range of benchmark problems, from sharp and moving features in two spatial dimensions to localised structures in up to eight spatial dimensions.

Problem

• high-dimensional PDEs
• physics-informed neural networks
• spacetime domains
• adaptive sampling
• PDE residual

Innovation

• Deep Adaptive Sampling
• Physics-Informed Neural Networks
• Normalising Flow
• High-Dimensional PDEs
• Spatiotemporal Domain