Modeling unsteady, fast transient, and advection-dominated physics problems is a pressing challenge for physics-aware deep learning (PADL). The physics of complex systems is governed by large systems of partial differential equations (PDEs) and ancillary constitutive models with nonlinear structures, as well as evolving state fields exhibiting sharp gradients and rapidly deforming material interfaces. Here, we investigate an inductive bias approach that is versatile and generalizable to model generic nonlinear field evolution problems. Our study focuses on the recent physics-aware recurrent convolutions (PARC), which incorporates a differentiator-integrator architecture that inductively models the spatiotemporal dynamics of generic physical systems. We extend the capabilities of PARC to simulate unsteady, transient, and advection-dominant systems. The extended model, referred to as PARCv2, is equipped with differential operators to model advection-reaction-diffusion equations, as well as a hybrid integral solver for stable, long-time predictions. PARCv2 is tested on both standard benchmark problems in fluid dynamics, namely Burgers and Navier-Stokes equations, and then applied to more complex shock-induced reaction problems in energetic materials. We evaluate the behavior of PARCv2 in comparison to other physics-informed and learning bias models and demonstrate its potential to model unsteady and advection-dominant dynamics regimes.

翻译：建模非稳态、快速瞬态及平流主导的物理问题是物理感知深度学习（PADL）面临的一项紧迫挑战。复杂系统的物理规律由大型偏微分方程（PDE）系统、具有非线性结构的辅助本构模型以及呈现陡峭梯度和快速变形材料界面的演化状态场所支配。本文研究一种适用于通用非线性场演化问题的、兼具灵活性与可推广性的归纳偏置方法。我们的研究聚焦于近期提出的物理感知循环卷积（PARC）方法，该方法通过微分-积分器架构归纳地建模通用物理系统的时空动力学。我们扩展了PARC的能力以模拟非稳态、瞬态及平流主导的系统。该扩展模型称为PARCv2，配备了用于建模平流-反应-扩散方程的微分算子，以及用于稳定长期预测的混合积分求解器。PARCv2在流体动力学标准基准问题（即Burgers方程和Navier-Stokes方程）上进行了测试，随后应用于含能材料中更复杂的冲击诱发反应问题。我们通过与其他物理信息模型及学习偏置模型的比较来评估PARCv2的性能，并证明其在建模非稳态与平流主导动力学体系方面的潜力。