# Accurate Data-Driven Surrogates of Dynamical Systems for Forward Propagation of Uncertainty

arXiv: https://arxiv.org/abs/2310.10831 · HTML: https://mlsanity.com/paper/2310.10831/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 49 references that are in the map.
- First posted on arXiv 16 Oct 2023.

## Builds on (most cited first, 5 of 5)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder](https://mlsanity.com/paper/2308.05882/) (2023)
- [Gradient-based Optimization for Regression in the Functional Tensor-Train Format](https://mlsanity.com/paper/1801.00885/) (2018)

## Built on by (most cited first, 1 of 1)

- [Improving the performance of Stein variational inference through extreme sparsification of physically-constrained neural network models](https://mlsanity.com/paper/2407.00761/) (2024)

## Abstract (arXiv)

Stochastic collocation (SC) is a well-known non-intrusive method of constructing surrogate models for uncertainty quantification. In dynamical systems, SC is especially suited for full-field uncertainty propagation that characterizes the distributions of the high-dimensional primary solution fields of a model with stochastic input parameters. However, due to the highly nonlinear nature of the parameter-to-solution map in even the simplest dynamical systems, the constructed SC surrogates are often inaccurate. This work presents an alternative approach, where we apply the SC approximation over the dynamics of the model, rather than the solution. By combining the data-driven sparse identification of nonlinear dynamics (SINDy) framework with SC, we construct dynamics surrogates and integrate them through time to construct the surrogate solutions. We demonstrate that the SC-over-dynamics framework leads to smaller errors, both in terms of the approximated system trajectories as well as the model state distributions, when compared against full-field SC applied to the solutions directly. We present numerical evidence of this improvement using three test problems: a chaotic ordinary differential equation, and two partial differential equations from solid mechanics.