Multi-fidelity physics constrained neural networks for dynamical systems
Hao Zhou, Sibo Cheng, Rossella Arcucci
TL;DR
The paper tackles the training cost and data efficiency challenges of physics-constrained neural networks for high-dimensional dynamical systems by introducing MSPCNN, which learns a shared latent space for high- and low-fidelity data via a dual autoencoder. An LSTM operating in this latent space predicts system evolution while physical constraints are enforced in the low-fidelity field, reducing computational load without sacrificing fidelity. Across Burgers' and shallow-water tests, MSPCNN achieves notable improvements in long-time accuracy and noise robustness, with substantial reductions in training time, especially when leveraging multiple constraints in the low-fidelity domain. The approach offers practical potential for real-time, multiscale CFD simulations and other multiscale physical systems, while acknowledging limitations such as potential error amplification and future work on more complex meshes and alternative architectures.
Abstract
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.