PARCv2: Physics-aware Recurrent Convolutional Neural Networks for Spatiotemporal Dynamics Modeling

TL;DR

PARCv2 extends physics-aware recurrent convolution networks (PARC) to advection–diffusion–reaction dynamics by incorporating spatial derivatives, a hybrid numerical/data-driven integration scheme, and a two-stage training pipeline. It learns a differentiator that approximates the PDE operators $F_x$ and $F_u$ and a data-driven integrator that corrects high-order errors, enabling stable, long-time predictions on benchmarks such as Burgers' equation, Navier–Stokes flows, and shock-induced hotspots in energetic materials. Results show PARCv2 achieving competitive RMSE and improved feature fidelity, while highlighting trade-offs with incompressibility in NS and the value of topological inductive bias for strongly nonlinear, advection-dominated regimes. The work suggests a complementary role for inductive-bias models alongside physics-informed losses and opens avenues for broader applications, including unknown-physics discovery and potential extensions to computer vision and stochastic dynamics.

Abstract

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.

Figures (6)

• Figure 1: PARCv2 architecture.
• Figure 2: Validation on the Burgers' equation.
• Figure 3: Prediction accuracy (left) and solution quality (right) on the Burgers' equation cases with respect to varying diffusion coefficients $R$.
• Figure 4: Validation on the Navier-Stokes equation (cylindrical flow).
• Figure 5: Prediction of energy localization in energetic materials.