# Physics guided neural networks for modelling of non-linear dynamics

arXiv: https://arxiv.org/abs/2205.06858 · HTML: https://mlsanity.com/paper/2205.06858/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 71 references that are in the map.
- First posted on arXiv 13 May 2022.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Algorithms for Verifying Deep Neural Networks](https://mlsanity.com/paper/1903.06758/) (2019)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications](https://mlsanity.com/paper/2010.02011/) (2020)
- [Physics guided machine learning using simplified theories](https://mlsanity.com/paper/2012.13343/) (2020)
- [Deep neural network enabled corrective source term approach to hybrid analysis and modeling](https://mlsanity.com/paper/2105.11521/) (2021)
- [Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution](https://mlsanity.com/paper/2103.14629/) (2021)
- [Physics-Incorporated Convolutional Recurrent Neural Networks for Source Identification and Forecasting of Dynamical Systems](https://mlsanity.com/paper/2004.06243/) (2020)
- [Taming an autonomous surface vehicle for path following and collision avoidance using deep reinforcement learning](https://mlsanity.com/paper/1912.08578/) (2019)

## Built on by (most cited first, 20 of 20)

- [State-space models are accurate and efficient neural operators for dynamical systems](https://mlsanity.com/paper/2409.03231/) (2024)
- [Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KAN](https://mlsanity.com/paper/2407.18373/) (2024)
- [Towards Complex Dynamic Physics System Simulation with Graph Neural ODEs](https://mlsanity.com/paper/2305.12334/) (2023)
- [Digital Twins in Wind Energy: Emerging Technologies and Industry-Informed Future Directions](https://mlsanity.com/paper/2304.11405/) (2023)
- [Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications](https://mlsanity.com/paper/2310.20425/) (2023)
- [Combining physics-based and data-driven techniques for reliable hybrid analysis and modeling using the corrective source term approach](https://mlsanity.com/paper/2206.03451/) (2022)
- [A novel corrective-source term approach to modeling unknown physics in aluminum extraction process](https://mlsanity.com/paper/2209.10861/) (2022)
- [Randomized prior wavelet neural operator for uncertainty quantification](https://mlsanity.com/paper/2302.01051/) (2023)
- [Decentralized digital twins of complex dynamical systems](https://mlsanity.com/paper/2207.12245/) (2022)
- [Point Neuron Learning: A New Physics-Informed Neural Network Architecture](https://mlsanity.com/paper/2408.16969/) (2024)
- [Machine Learning for Smart and Energy-Efficient Buildings](https://mlsanity.com/paper/2211.14889/) (2022)
- [Unified Spatiotemporal Physics-Informed Learning (USPIL): A Framework for Modeling Complex Predator-Prey Dynamics](https://mlsanity.com/paper/2509.13425/) (2025)
- [A novel framework for generalization of deep hidden physics models](https://mlsanity.com/paper/2401.04648/) (2024)
- [Aeroengine performance prediction using a physical-embedded data-driven method](https://mlsanity.com/paper/2407.00501/) (2024)
- [Enhanced accuracy through ensembling of randomly initialized auto-regressive models for time-dependent PDEs](https://mlsanity.com/paper/2507.03863/) (2025)
- [Learning with Mandelbrot and Julia](https://mlsanity.com/paper/2509.00903/) (2025)
- [Enabling Physical AI at the Edge: Hardware-Accelerated Recovery of System Dynamics](https://mlsanity.com/paper/2512.23767/) (2025)
- [Hamiltonian Graph Inference Networks: Joint structure discovery and dynamics prediction for lattice Hamiltonian systems from trajectory data](https://mlsanity.com/paper/2604.23606/) (2026)
- [Geometric Kolmogorov--Arnold Network (GeoKAN)](https://mlsanity.com/paper/2605.06740/) (2026)
- [Neural Operator based Multi-Field Reconstruction of Inner Solar Boundary State](https://mlsanity.com/paper/2608.22782/) (2026)

## Abstract (arXiv)

The success of the current wave of artificial intelligence can be partly attributed to deep neural networks, which have proven to be very effective in learning complex patterns from large datasets with minimal human intervention. However, it is difficult to train these models on complex dynamical systems from data alone due to their low data efficiency and sensitivity to hyperparameters and initialisation. This work demonstrates that injection of partially known information at an intermediate layer in a DNN can improve model accuracy, reduce model uncertainty, and yield improved convergence during the training. The value of these physics-guided neural networks has been demonstrated by learning the dynamics of a wide variety of nonlinear dynamical systems represented by five well-known equations in nonlinear systems theory: the Lotka-Volterra, Duffing, Van der Pol, Lorenz, and Henon-Heiles systems.