# On instabilities in neural network-based physics simulators

arXiv: https://arxiv.org/abs/2406.13101 · HTML: https://mlsanity.com/paper/2406.13101/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 32 references that are in the map.
- First posted on arXiv 18 Jun 2024.

## Builds on (most cited first, 18 of 18)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Forecasting Global Weather with Graph Neural Networks](https://mlsanity.com/paper/2202.07575/) (2022)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamics](https://mlsanity.com/paper/1910.05266/) (2019)
- [Multiscale Simulations of Complex Systems by Learning their Effective Dynamics](https://mlsanity.com/paper/2006.13431/) (2020)
- [Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems](https://mlsanity.com/paper/2203.15706/) (2022)
- [Multi-stage Neural Networks: Function Approximator of Machine Precision](https://mlsanity.com/paper/2307.08934/) (2023)
- [Data-driven discovery of intrinsic dynamics](https://mlsanity.com/paper/2108.05928/) (2021)
- [Deep learning to discover and predict dynamics on an inertial manifold](https://mlsanity.com/paper/2001.04263/) (2019)
- [Challenges of learning multi-scale dynamics with AI weather models: Implications for stability and one solution](https://mlsanity.com/paper/2304.07029/) (2023)
- [Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2109.00060/) (2021)
- [Parallel Machine Learning for Forecasting the Dynamics of Complex Networks](https://mlsanity.com/paper/2108.12129/) (2021)

## Built on by (most cited first, 2 of 2)

- [No Free Lunch in Flow Surrogates under Time-Varying Boundary Conditions: A Two-Regime Study](https://mlsanity.com/paper/2607.23667/) (2026)
- [Eigenanalysis framework for autoregressive neural emulators of multi-scale chaotic dynamics](https://mlsanity.com/paper/2608.16084/) (2026)

## Abstract (arXiv)

When neural networks are trained from data to simulate the dynamics of physical systems, they encounter a persistent challenge: the long-time dynamics they produce are often unphysical or unstable. We analyze the origin of such instabilities when learning linear dynamical systems, focusing on the training dynamics. We make several analytical findings which empirical observations suggest extend to nonlinear dynamical systems. First, the rate of convergence of the training dynamics is uneven and depends on the distribution of energy in the data. As a special case, the dynamics in directions where the data have no energy cannot be learned. Second, in the unlearnable directions, the dynamics produced by the neural network depend on the weight initialization, and common weight initialization schemes can produce unstable dynamics. Third, injecting synthetic noise into the data during training adds damping to the training dynamics and can stabilize the learned simulator, though doing so undesirably biases the learned dynamics. For each contributor to instability, we suggest mitigative strategies. We also highlight important differences between learning discrete-time and continuous-time dynamics, and discuss extensions to nonlinear systems.