Train Small, Model Big: Scalable Physics Simulators via Reduced Order Modeling and Domain Decomposition

TL;DR

This work tackles the challenge of scaling lab-scale physics to industrial scales by marrying physics-conscious reduced-order modeling with discontinuous Galerkin domain decomposition. By training ROM components on small unit cells and stitching them via DG-DD, the authors construct global simulations that extrapolate robustly to industrial scales. The Poisson and Stokes examples show $15$–$40$× speedups, around $1\%$ relative error, and memory reductions of about an order of magnitude, underscoring practical impact for large-scale engineering prediction and optimization. The approach is poised to enable faster design, testing, and deployment of complex multiphysics systems, with clear paths to nonlinear, three-dimensional, and more heterogeneous applications.

Abstract

Numerous cutting-edge scientific technologies originate at the laboratory scale, but transitioning them to practical industry applications is a formidable challenge. Traditional pilot projects at intermediate scales are costly and time-consuming. An alternative, the E-pilot, relies on high-fidelity numerical simulations, but even these simulations can be computationally prohibitive at larger scales. To overcome these limitations, we propose a scalable, physics-constrained reduced order model (ROM) method. ROM identifies critical physics modes from small-scale unit components, projecting governing equations onto these modes to create a reduced model that retains essential physics details. We also employ Discontinuous Galerkin Domain Decomposition (DG-DD) to apply ROM to unit components and interfaces, enabling the construction of large-scale global systems without data at such large scales. This method is demonstrated on the Poisson and Stokes flow equations, showing that it can solve equations about $15 - 40$ times faster with only $\sim$ $1\%$ relative error. Furthermore, ROM takes one order of magnitude less memory than the full order model, enabling larger scale predictions at a given memory limitation.

Figures (12)

• Figure 1: Illustration of domain decomposition.
• Figure 2: Diagram of the component reduced-order modeling procedure.
• Figure 3: Sample generation and linear subspace training for the Poisson equation: (a) a sample solution in the component domain $\overline{\Omega}_1$ with $\mathbf{k} = (0.4, -0.3)$, $\mathbf{k}_b = (0.5, 0.4)$, $\theta=0.1$ and $\theta_b=0.4$; and (b) Singular value spectrum of POD basis identified from $\mathbf{Q}_1$. The gray line indicates the $15$-th singular value, up to which $99.77\%$ of spectrum is covered.
• Figure 4: Example predictions on the global domain $\Omega=[0,32]^2$ with the global ROM (\ref{['eq:weak-gov-reduced']}) constructed with $32^2$ components: ROM solutions for (a) the component problem (\ref{['eq:poisson-component']}) with $\mathbf{k}=(0.6, -0.6)$, $\mathbf{k}_b=(0.05, 0.04)$, $\theta=0.1$ and $\theta_b=0$; and for (b) the Spiral problem (\ref{['eq:poisson-spiral']}) with $(s, k) = (1.53, 3)$. The component-level sample from Figure \ref{['fig:poisson-sample']} is included as a reference to compare size.
• Figure 5: Performance of the ROM compared to the FOM for the Poisson equation, depending on the size of the domain: (a) assembly time of the system; (b) computation time of the system; and (c) relative error of the ROM compared to the FOM solution. The marker denotes the median value of 100 test cases, and the error bar denotes $95\%$-confidence interval of the test cases. Both ROM and FOM systems are solved using the conjugate-gradient method without any preconditioner.