# Physically Informed Synchronic-adaptive Learning for Industrial Systems Modeling in Heterogeneous Media with Unavailable Time-varying Interface

arXiv: https://arxiv.org/abs/2401.14609 · HTML: https://mlsanity.com/paper/2401.14609/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 61 references that are in the map.
- First posted on arXiv 26 Jan 2024.

## Builds on (most cited first, 10 of 10)

- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutions](https://mlsanity.com/paper/2212.07564/) (2022)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [Deep learning of free boundary and Stefan problems](https://mlsanity.com/paper/2006.05311/) (2020)
- [Physics-constrained Deep Learning for Robust Inverse ECG Modeling](https://mlsanity.com/paper/2107.12780/) (2021)
- [The Stefan problem and free targets of optimal Brownian martingale transport](https://mlsanity.com/paper/2110.03831/) (2021)
- [Solving PDEs with Unmeasurable Source Terms Using Coupled Physics-Informed Neural Network with Recurrent Prediction for Soft Sensors](https://mlsanity.com/paper/2301.08618/) (2023)

## Abstract (arXiv)

Partial differential equations (PDEs) are commonly employed to model complex industrial systems characterized by multivariable dependence. Existing physics-informed neural networks (PINNs) excel in solving PDEs in a homogeneous medium. However, their feasibility is diminished when PDE parameters are unknown due to a lack of physical attributions and time-varying interface is unavailable arising from heterogeneous media. To this end, we propose a data-physics-hybrid method, physically informed synchronic-adaptive learning (PISAL), to solve PDEs for industrial systems modeling in heterogeneous media. First, Net1, Net2, and NetI, are constructed to approximate the solutions satisfying PDEs and the interface. Net1 and Net2 are utilized to synchronously learn each solution satisfying PDEs with diverse parameters, while NetI is employed to adaptively learn the unavailable time-varying interface. Then, a criterion combined with NetI is introduced to adaptively distinguish the attributions of measurements and collocation points. Furthermore, NetI is integrated into a data-physics-hybrid loss function. Accordingly, a synchronic-adaptive learning (SAL) strategy is proposed to decompose and optimize each subdomain. Besides, we theoretically prove the approximation capability of PISAL. Extensive experimental results verify that the proposed PISAL can be used for industrial systems modeling in heterogeneous media, which faces the challenges of lack of physical attributions and unavailable time-varying interface.