# A Control Perspective on Training PINNs

arXiv: https://arxiv.org/abs/2501.18582 · HTML: https://mlsanity.com/paper/2501.18582/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 36 references that are in the map.
- First posted on arXiv 30 Jan 2025.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [Lagrangian Duality for Constrained Deep Learning](https://mlsanity.com/paper/2001.09394/) (2020)
- [Meta-learning PINN loss functions](https://mlsanity.com/paper/2107.05544/) (2021)
- [ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2408.11104/) (2024)
- [Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2107.00940/) (2021)
- [Continuous vs. Discrete Optimization of Deep Neural Networks](https://mlsanity.com/paper/2107.06608/) (2021)
- [On the Exponential Stability of Primal-Dual Gradient Dynamics](https://mlsanity.com/paper/1803.01825/) (2018)
- [A new framework for constrained optimization via feedback control of Lagrange multipliers](https://mlsanity.com/paper/2403.12738/) (2024)
- [Robust Decentralized Secondary Frequency Control in Power Systems: Merits and Trade-Offs](https://mlsanity.com/paper/1711.07332/) (2017)
- [MILP initialization for solving parabolic PDEs with PINNs](https://mlsanity.com/paper/2501.16153/) (2025)

## Built on by (most cited first, 3 of 3)

- [Iterative Training of Physics-Informed Neural Networks with Fourier-enhanced Features](https://mlsanity.com/paper/2510.19399/) (2025)
- [Data-driven multi-agent modelling of calcium interactions in cell culture: PINN vs Regularized Least-squares](https://mlsanity.com/paper/2505.20327/) (2025)
- [Curriculum-Learned Vanishing Stacked Residual PINNs for Hyperbolic PDE State Reconstruction](https://mlsanity.com/paper/2602.06996/) (2026)

## Abstract (arXiv)

We investigate the training of Physics-Informed Neural Networks (PINNs) from a control-theoretic perspective. Using gradient descent with resampling, we interpret the training dynamics as asymptotically equivalent to a stochastic control-affine system, where sampling effects act as process disturbances and measurement noise. Within this framework, we introduce two controllers for dynamically adapting the physics weight: an integral controller and a leaky integral controller. We theoretically analyze their asymptotic properties under the accuracy-robustness trade-off, and we evaluate them on a toy example. Numerical evidence suggests that the integral controller achieves accurate and robust convergence when the physical model is correct, whereas the leaky integrator provides improved performance in the presence of model mismatch. This work represents a first step toward convergence guarantees and principled training algorithms tailored to the distinct characteristics of PINN tasks.