Comparative Study of Adjoint Methods and Physics-Informed Neural Networks for Solving Inverse PDE Problems
Researchers conducted a systematic comparison of adjoint-based optimization and physics-informed neural networks (PINNs) for solving inverse problems governed by partial differential equations, using identical formulations and parameters. The study found that the choice of method depends on how the unknown is represented: grid-based fields favor traditional adjoint methods, while neural representations suit PINNs better. The findings suggest a hybrid approach combining both methods can achieve high accuracy at reduced computational cost.
A new preprint from arXiv presents a rigorous benchmarking study comparing two major approaches to PDE-constrained inverse problems: classical adjoint-based optimization and the emerging physics-informed neural networks (PINNs) methodology. To ensure fairness, the researchers implemented both methods using identical problem formulations, domains, governing equations, observation models, regularization schemes, optimizers, and numerical precision across four test cases: unsteady Burgers equation, noisy Darcy permeability inversion, three-dimensional Allen-Cahn reaction identification, and unsteady Navier-Stokes viscosity identification. The results reveal that representation choice is critical: discrete adjoint methods excel when unknowns are represented on grids, while PINNs naturally handle neural representations and are particularly advantageous for time-dependent problems where adjoint methods suffer from high memory costs for trajectory storage. The authors propose a hybrid strategy where PINNs warm-start the adjoint solver, achieving adjoint-level accuracy with substantially lower computational overhead.
What's missing
The study does not discuss computational wall-clock time comparisons or provide guidance on how practitioners should choose between methods for specific real-world applications beyond the tested benchmarks. Additionally, the scalability of both approaches to very high-dimensional problems or to PDEs with complex nonlinearities beyond those tested is not addressed.
What different sources said
- arXiv cs.LGCenter
Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems
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