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

Problem
The paper addresses the lack of a standardized comparison between adjoint-based optimization and physics-informed neural networks (PINNs) in the context of PDE-constrained inverse problems. Previous studies often employed disparate formulations, parameterizations, and optimization strategies, making it challenging to assess the relative performance of these two methodologies. This work aims to provide a fair and systematic evaluation of both approaches under identical conditions, which is particularly relevant given the increasing adoption of PINNs in computational mechanics.

Method
The authors establish a common abstract formulation for both adjoint optimization and PINNs, ensuring that they are instantiated on the same domains, governing equations, observation models, and regularization terms. They match the optimizer, parameterization of unknowns, and arithmetic precision across both methods. The benchmarks utilized include unsteady Burgers’ equation, noisy Darcy permeability inversion, three-dimensional Allen–Cahn reaction identification, and unsteady Navier–Stokes viscosity identification. The study employs a variety of metrics to evaluate performance, including reconstruction accuracy and computational cost, with a specific focus on the representation of unknown parameters. Additionally, a novel PINN-warm-started adjoint strategy is introduced to enhance efficiency while maintaining accuracy.

Results
The findings indicate that the choice of method is significantly influenced by the representation of the unknowns. For grid-based fields, the discrete adjoint method outperforms PINNs, while for neural representations, PINNs demonstrate superior performance, particularly in closure and constitutive modeling. In time-dependent scenarios, adjoint inversion is often hindered by the computational burden associated with trajectory storage and differentiation, whereas PINNs achieve satisfactory reconstructions at a lower computational cost. The proposed PINN-warm-started adjoint strategy successfully recovers adjoint-level accuracy with a substantial reduction in computational expense, showcasing a potential hybrid approach that leverages the strengths of both methodologies.

Limitations
The authors acknowledge that their comparison is limited to the specific PDEs and formulations chosen for the benchmarks, which may not generalize to all inverse problems. They also note that the performance of the PINN-warm-started adjoint strategy may vary depending on the complexity of the problem and the choice of hyperparameters. Additionally, the study does not explore the impact of different optimizers or regularization techniques beyond those employed in the experiments, which could influence the results.

Why it matters
This work provides critical insights into the comparative strengths and weaknesses of adjoint optimization and PINNs, informing future research and applications in computational mechanics and inverse problems. The findings suggest that the choice of method should be guided by the nature of the unknowns and the specific problem context, which can lead to more efficient and accurate solutions. This comparative framework can serve as a foundation for further exploration of hybrid approaches, as discussed in the context of advancing methodologies in the field, as published in arXiv cs.LG.