# Multi-fidelity physics constrained neural networks for dynamical systems

arXiv: https://arxiv.org/abs/2402.02031 · HTML: https://mlsanity.com/paper/2402.02031/

## Key facts

- Cited by 5 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 3 Feb 2024.

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

- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Deep Fluids: A Generative Network for Parameterized Fluid Simulations](https://mlsanity.com/paper/1806.02071/) (2018)
- [Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks](https://mlsanity.com/paper/1905.04817/) (2019)
- [Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction](https://mlsanity.com/paper/1905.10866/) (2019)
- [Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [Generalised Latent Assimilation in Heterogeneous Reduced Spaces with Machine Learning Surrogate Models](https://mlsanity.com/paper/2204.03497/) (2022)
- [Multi-fidelity Generative Deep Learning Turbulent Flows](https://mlsanity.com/paper/2006.04731/) (2020)
- [Multi-fidelity surrogate modeling using long short-term memory networks](https://mlsanity.com/paper/2208.03115/) (2022)
- [Multi-fidelity reduced-order surrogate modeling](https://mlsanity.com/paper/2309.00325/) (2023)
- [Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models](https://mlsanity.com/paper/2106.05722/) (2021)
- [Latent-space time evolution of non-intrusive reduced-order models using Gaussian process emulation](https://mlsanity.com/paper/2007.12167/) (2020)
- [Background Error Covariance Iterative Updating with Invariant Observation Measures for Data Assimilation](https://mlsanity.com/paper/1910.09408/) (2019)
- [textitFastSVD-ML-ROM: A Reduced-Order Modeling Framework based on Machine Learning for Real-Time Applications](https://mlsanity.com/paper/2207.11842/) (2022)

## Built on by (most cited first, 5 of 5)

- [Dynamical system prediction from sparse observations using deep neural networks with Voronoi tessellation and physics constraint](https://mlsanity.com/paper/2409.00458/) (2024)
- [Improving Long-term Autoregressive Spatiotemporal Predictions: A Proof of Concept with Fluid Dynamics](https://mlsanity.com/paper/2508.18565/) (2025)
- [Time Extrapolation with Graph Convolutional Autoencoder and Tensor Train Decomposition](https://mlsanity.com/paper/2511.23037/) (2025)
- [Artificial Intelligence in Reactor Physics: Current Status and Future Prospects](https://mlsanity.com/paper/2503.02440/) (2025)
- [Sparsely-Supervised Data Assimilation via Physics-Informed Schr"odinger Bridge](https://mlsanity.com/paper/2603.22319/) (2026)

## Abstract (arXiv)

Physics-constrained neural networks are commonly employed to enhance prediction robustness compared to purely data-driven models, achieved through the inclusion of physical constraint losses during the model training process. However, one of the major challenges of physics-constrained neural networks consists of the training complexity especially for high-dimensional systems. In fact, conventional physics-constrained models rely on singular-fidelity data necessitating the assessment of physical constraints within high-dimensional fields, which introduces computational difficulties. Furthermore, due to the fixed input size of the neural networks, employing multi-fidelity training data can also be cumbersome. In this paper, we propose the Multi-Scale Physics-Constrained Neural Network (MSPCNN), which offers a novel methodology for incorporating data with different levels of fidelity into a unified latent space through a customised multi-fidelity autoencoder. Additionally, multiple decoders are concurrently trained to map latent representations of inputs into various fidelity physical spaces. As a result, during the training of predictive models, physical constraints can be evaluated within low-fidelity spaces, yielding a trade-off between training efficiency and accuracy. In addition, unlike conventional methods, MSPCNN also manages to employ multi-fidelity data to train the predictive model. We assess the performance of MSPCNN in two fluid dynamics problems, namely a two-dimensional Burgers' system and a shallow water system. Numerical results clearly demonstrate the enhancement of prediction accuracy and noise robustness when introducing physical constraints in low-fidelity fields. On the other hand, as expected, the training complexity can be significantly reduced by computing physical constraint loss in the low-fidelity field rather than the high-fidelity one.