Meshless solutions of PDE inverse problems on irregular geometries

2025-10-29

AI Summary

Solving inverse problems and coupled optimization tasks for nonlinear partial differential equations (PDEs) on complex geometric domains remains challenging due to meshing difficulties, discretization errors, and slow convergence. Method: We propose a mesh-free spectral physics-informed method that constructs parameterized spectral basis functions on an enclosing hyperrectangle and enforces PDE constraints, boundary conditions, and optimization objectives through a unified physics-informed loss function. Contribution/Results: This work is the first to tightly integrate globally exponential-convergent spectral approximation with the physics-informed neural network paradigm—eliminating mesh generation and associated discretization errors while enabling data assimilation and end-to-end optimization. Numerical experiments demonstrate exponential convergence of solutions across diverse nonlinear PDE inverse problems, significantly improving both accuracy and efficiency on irregular domains. The framework provides a general, efficient, and truly mesh-free approach for inverse modeling of PDEs on complex geometries.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems

Abstract

Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral bases on arbitrary spatiotemporal domains, whereby the basis is defined on a hyperrectangle containing the true domain. We find the coefficients of the basis expansion by solving an optimization problem whereby both the equations, the boundary conditions and any optimization targets are enforced by a loss function, building on a key idea from Physics-Informed Neural Networks (PINNs). Since the representation of the function natively has exponential convergence, so does the solution of the optimization problem, as long as it can be solved efficiently. We find empirically that the optimization protocols developed for machine learning find solutions with exponential convergence on a wide range of equations. The method naturally allows for the incorporation of data assimilation by including additional terms in the loss function, and for the efficient solution of optimization problems over the PDE solutions.

Problem

• Solving PDE inverse problems on complex irregular geometries efficiently
• Developing meshless methods with exponential convergence for nonlinear PDEs
• Incorporating data assimilation and optimization into PDE solution frameworks

Innovation

• Spectral bases parameterize PDE solutions on irregular domains
• Loss function enforces equations and boundary conditions
• Method achieves exponential convergence in optimization problems