# Physics-Informed Neural Nets for Control of Dynamical Systems

arXiv: https://arxiv.org/abs/2104.02556 · HTML: https://mlsanity.com/paper/2104.02556/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 63 references that are in the map.
- First posted on arXiv 6 Apr 2021.

## Builds on (most cited first, 13 of 13)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Deeply-Supervised Nets](https://mlsanity.com/paper/1409.5185/) (2014)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations](https://mlsanity.com/paper/1907.03507/) (2019)
- [Learning an Approximate Model Predictive Controller with Guarantees](https://mlsanity.com/paper/1806.04167/) (2018)
- [Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations](https://mlsanity.com/paper/1804.07010/) (2018)
- [Enforcing constraints for time series prediction in supervised, unsupervised and reinforcement learning](https://mlsanity.com/paper/1905.07501/) (2019)

## Built on by (most cited first, 30 of 31)

- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [LyaNet: A Lyapunov Framework for Training Neural ODEs](https://mlsanity.com/paper/2202.02526/) (2022)
- [Physics-informed neural networks for PDE-constrained optimization and control](https://mlsanity.com/paper/2205.03377/) (2022)
- [RAMP-Net: A Robust Adaptive MPC for Quadrotors via Physics-informed Neural Network](https://mlsanity.com/paper/2209.09025/) (2022)
- [Deep Operator Networks for Bayesian Parameter Estimation in PDEs](https://mlsanity.com/paper/2501.10684/) (2025)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [Physics-Informed Neural Networks with Skip Connections for Modeling and Control of Gas-Lifted Oil Wells](https://mlsanity.com/paper/2403.02289/) (2024)
- [Domain-decoupled Physics-informed Neural Networks with Closed-form Gradients for Fast Model Learning of Dynamical Systems](https://mlsanity.com/paper/2408.14951/) (2024)
- [Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order models](https://mlsanity.com/paper/2409.05709/) (2024)
- [Generalizable and Fast Surrogates: Model Predictive Control of Articulated Soft Robots using Physics-Informed Neural Networks](https://mlsanity.com/paper/2502.01916/) (2025)
- [Adaptive Model-Predictive Control of a Soft Continuum Robot Using a Physics-Informed Neural Network Based on Cosserat Rod Theory](https://mlsanity.com/paper/2508.12681/) (2025)
- [Actor-Critic Methods using Physics-Informed Neural Networks: Control of a 1D PDE Model for Fluid-Cooled Battery Packs](https://mlsanity.com/paper/2305.10952/) (2023)
- [Common pitfalls to avoid while using multiobjective optimization in machine learning](https://mlsanity.com/paper/2405.01480/) (2024)
- [Gradient Flow Based Phase-Field Modeling Using Separable Neural Networks](https://mlsanity.com/paper/2405.06119/) (2024)
- [Solving Differential Equations using Physics-Informed Deep Equilibrium Models](https://mlsanity.com/paper/2406.03472/) (2024)
- [Physics-Informed Machine Learning for Grade Prediction in Froth Flotation](https://mlsanity.com/paper/2408.15267/) (2024)
- [Robust Neural IDA-PBC: passivity-based stabilization under approximations](https://mlsanity.com/paper/2409.16008/) (2024)
- [MILP initialization for solving parabolic PDEs with PINNs](https://mlsanity.com/paper/2501.16153/) (2025)
- [Data-Driven Adaptive PID Control Based on Physics-Informed Neural Networks](https://mlsanity.com/paper/2510.04591/) (2025)
- [Tensor-Compressed and Fully-Quantized Training of Neural PDE Solvers](https://mlsanity.com/paper/2512.09202/) (2025)
- [Receding Hamiltonian-Informed Optimal Neural Control and State Estimation for Closed-Loop Dynamical Systems](https://mlsanity.com/paper/2411.01297/) (2024)
- [Latent feedback control of distributed systems in multiple scenarios through deep learning-based reduced order models](https://mlsanity.com/paper/2412.09942/) (2024)
- [Acoustic Wave Manipulation Through Sparse Robotic Actuation](https://mlsanity.com/paper/2502.08784/) (2025)
- [A Low-complexity Structured Neural Network to Realize States of Dynamical Systems](https://mlsanity.com/paper/2503.23697/) (2025)
- [Modelling of Underwater Vehicles using Physics-Informed Neural Networks with Control](https://mlsanity.com/paper/2504.20019/) (2025)
- [Physics-Informed Neural Networks for Control of Single-Phase Flow Systems Governed by Partial Differential Equations](https://mlsanity.com/paper/2506.06188/) (2025)
- [Learning Robust Satellite Attitude Dynamics with Physics-Informed Normalising Flow](https://mlsanity.com/paper/2508.07841/) (2025)
- [PCARNN-DCBF: Minimal-Intervention Geofence Enforcement for Ground Vehicles](https://mlsanity.com/paper/2511.15522/) (2025)
- [Learned Lyapunov Shielding for Adaptive Control](https://mlsanity.com/paper/2605.06934/) (2026)
- [Hierarchical Causal Abduction: A Foundation Framework for Explainable Model Predictive Control](https://mlsanity.com/paper/2605.10624/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) impose known physical laws into the learning of deep neural networks, making sure they respect the physics of the process while decreasing the demand of labeled data. For systems represented by Ordinary Differential Equations (ODEs), the conventional PINN has a continuous time input variable and outputs the solution of the corresponding ODE. In their original form, PINNs do not allow control inputs, neither can they simulate for variable long-range intervals without serious degradation in their predictions. In this context, this work presents a new framework called Physics-Informed Neural Nets for Control (PINC), which proposes a novel PINN-based architecture that is amenable to control problems and able to simulate for longer-range time horizons that are not fixed beforehand, making it a very flexible framework when compared to traditional PINNs. Furthermore, this long-range time simulation of differential equations is faster than numerical methods since it relies only on signal propagation through the network, making it less computationally costly and, thus, a better alternative for simulation of models in Model Predictive Control. We showcase our proposal in the control of two nonlinear dynamic systems: the Van der Pol oscillator and the four-tank system.