Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects

2026-05-16

AI Summary

This work addresses inverse problems governed by partial differential equations (inverse PDEs), which aim to infer hidden physical mechanisms, structural designs, or system controls from observational data, with broad applications in medical imaging, geophysics, and materials science. The study proposes a unified AI-driven framework that categorizes inverse PDE problems into three classes: inverse inference, inverse design, and control. It systematically reviews integrative methodologies, including physics-informed neural networks, deep generative models, optimization-driven learning, and data assimilation. Beyond summarizing representative applications in full-waveform inversion, thermal systems, and mechanical systems, the paper further identifies emerging frontiers such as physics-informed architectures, few-shot learning, uncertainty quantification, and foundational inverse models, highlighting the transformative potential of artificial intelligence in reshaping traditional paradigms for solving inverse problems.

Abstract

Solving inverse partial differential equation (PDE) problems is a fundamental topic in scientific research due to its broad significance across a wide range of real-world applications. Inverse PDE problems arise across medical imaging, geophysics, materials science, and aerodynamics, where the goal is to infer hidden causes, design structures, or control physical states. In this paper, we provide a comprehensive review of recent advances in solving inverse PDE problems using artificial intelligence (AI). We first introduce the basic formulation, key challenges, and traditional numerical foundations of inverse PDE problems, and then organize it into three major categories: inverse problems, inverse design, and control problems. For each category, we further present a methodological paradigms, and review representative state-of-the-art approaches from recent years. We then summarize representative applications across scientific and industrial domains, including mechanical systems, aerodynamic problems, thermal systems, full-waveform inversion, system identification, and medical imaging. Finally, we discuss open challenges and future prospects, such as physics-informed architectures, limited real-world data, uncertainty quantification, and inverse foundation models. This survey aims to provide the first unified and systematic perspective on AI for inverse PDE problems, demonstrating how modern learning-based methods are reshaping inverse problems, inverse design, and control problems in PDE-governed systems.

Problem

• inverse PDE
• artificial intelligence
• scientific computing
• inverse problems
• physics-informed learning

Innovation

• inverse PDE
• artificial intelligence
• physics-informed learning
• inverse design
• uncertainty quantification