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

arXiv: https://arxiv.org/abs/2606.12337 · HTML: https://mlsanity.com/paper/2606.12337/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 29 references that are in the map.
- First posted on arXiv 10 Jun 2026.

## Builds on (most cited first, 16 of 16)

- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [SOAP: Improving and Stabilizing Shampoo using Adam](https://mlsanity.com/paper/2409.11321/) (2024)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2501.16371/) (2025)
- [Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?](https://mlsanity.com/paper/2405.04230/) (2024)
- [Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks](https://mlsanity.com/paper/2604.05230/) (2026)
- [A Comparison of Mesh-Free Differentiable Programming and Data-Driven Strategies for Optimal Control under PDE Constraints](https://mlsanity.com/paper/2310.02286/) (2023)
- [Online identification and control of PDEs via Reinforcement Learning methods](https://mlsanity.com/paper/2308.04068/) (2023)
- [A PINN approach for the online identification and control of unknown PDEs](https://mlsanity.com/paper/2408.03456/) (2024)
- [PINNs in PDE Constrained Optimal Control Problems: Direct vs Indirect Methods](https://mlsanity.com/paper/2604.04920/) (2026)
- [A POD approach to identify and control PDEs online through State Dependent Riccati equations](https://mlsanity.com/paper/2402.08186/) (2024)
- [Self-Scaled Broyden Family of Quasi-Newton Methods in JAX](https://mlsanity.com/paper/2603.10599/) (2026)
- [A parallel-in-time approach for accelerating direct-adjoint studies](https://mlsanity.com/paper/2004.00546/) (2020)

## Abstract (arXiv)

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 controlled 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 about half the cost.