Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems

2026-06-10

AI Summary

This study addresses the lack of fair comparison between adjoint-based optimization and physics-informed neural networks (PINNs) in solving PDE-constrained inverse problems by conducting a systematic evaluation within a unified framework, ensuring identical settings for governing equations, observation models, regularization, parameterization, and optimizers. The analysis reveals that the representation of unknown fields critically determines method performance: grid-discretized fields favor adjoint methods, whereas neural network representations align better with PINNs. Building on this insight, the authors propose a hybrid strategy that uses PINN-based pre-training to initialize adjoint optimization, achieving high accuracy while substantially reducing computational cost. This work presents the first equitable benchmark between these two dominant approaches and establishes a new paradigm for efficiently solving high-dimensional spatiotemporal inverse problems.

Abstract

Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative. Their relative performance remains difficult to assess because the two approaches are often compared under different formulations, parameterizations, optimizers, and regularization choices. We present a fair comparison of adjoint optimization and PINNs for PDE-constrained inverse problems. From a common abstract formulation, we instantiate both methods on identical domains, governing equations, observation models, and regularization terms, while matching the optimizer, unknown parameterization, and arithmetic precision wherever applicable. The benchmarks include unsteady Burgers, noisy Darcy permeability inversion, three-dimensional Allen--Cahn reaction identification, and unsteady Navier--Stokes viscosity identification. The results show that the representation of the unknown largely determines the preferred method: grid-based fields favor the discrete adjoint, whereas neural representations are native to PINNs and relevant for closure and constitutive modeling. For time-dependent problems, adjoint inversion can be dominated by trajectory storage and differentiation, while PINNs provide satisfactory reconstructions at lower cost. A PINN-warm-started adjoint strategy then recovers adjoint-level accuracy at substantially reduced cost.

Problem

• PDE-constrained inverse problems
• adjoint method
• physics-informed neural networks
• fair comparison
• parameterization

Innovation

• Adjoint Method
• Physics-Informed Neural Networks
• PDE-Constrained Inverse Problems
• Fair Benchmarking
• Hybrid Optimization Strategy