# MCMC Informed Neural Emulators for Uncertainty Quantification in Dynamical Systems

arXiv: https://arxiv.org/abs/2603.10987 · HTML: https://mlsanity.com/paper/2603.10987/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 46 references that are in the map.
- First posted on arXiv 11 Mar 2026.

## Builds on (most cited first, 19 of 19)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems](https://mlsanity.com/paper/2203.15706/) (2022)
- [Learning Stochastic Dynamics with Statistics-Informed Neural Network](https://mlsanity.com/paper/2202.12278/) (2022)
- [A practical PINN framework for multi-scale problems with multi-magnitude loss terms](https://mlsanity.com/paper/2308.06672/) (2023)
- [Hybrid2 Neural ODE Causal Modeling and an Application to Glycemic Response](https://mlsanity.com/paper/2402.17233/) (2024)
- [FunnyNet-W: Multimodal Learning of Funny Moments in Videos in the Wild](https://mlsanity.com/paper/2401.04210/) (2024)
- [From Fourier to Neural ODEs: Flow Matching for Modeling Complex Systems](https://mlsanity.com/paper/2405.11542/) (2024)
- [Speeding up astrochemical reaction networks with autoencoders and neural ODEs](https://mlsanity.com/paper/2312.06015/) (2023)
- [Neural Network Emulator for Atmospheric Chemical ODE](https://mlsanity.com/paper/2408.01829/) (2024)
- [Parareal with a physics-informed neural network as coarse propagator](https://mlsanity.com/paper/2303.03848/) (2023)
- [Multiscale Corrections by Continuous Super-Resolution](https://mlsanity.com/paper/2411.07576/) (2024)
- [Adaptive Movement Sampling Physics-Informed Residual Network (AM-PIRN) for Solving Nonlinear Option Pricing models](https://mlsanity.com/paper/2504.03244/) (2025)

## Abstract (arXiv)

Neural networks are a commonly used approach to replace physical models with computationally cheap surrogates. Parametric uncertainty quantification can be included in training, assuming that an accurate prior distribution of the model parameters is available. Here we study the common opposite situation, where direct screening or random sampling of model parameters leads to exhaustive training times and evaluations at unphysical parameter values. Our solution is to decouple uncertainty quantification from network architecture. Instead of sampling network weights, we introduce the model-parameter distribution as an input to network training via Markov chain Monte Carlo (MCMC). In this way, the surrogate achieves the same uncertainty quantification as the underlying physical model, but with substantially reduced computation time. The approach is fully agnostic with respect to the neural network choice. In our examples, we present a quantile emulator for prediction and a novel autoencoder-based ODE network emulator that can flexibly estimate different trajectory paths corresponding to different ODE model parameters. Moreover, we present a mathematical analysis that provides a transparent way to relate potential performance loss to measurable distribution mismatch.