Title: Multiscaling and Machine Learning Approaches to Physics Simulation
Description: Physics simulation computationally models physical phenomena. It is the bread-and-butter of modern-day scientific discoveries and engineering design: from plasma theory to digital twins. However, viable efficiency remains a long-standing challenge to physics simulation. Accurate, real-world-scale simulations are often computationally too expensive (e.g., excessive wall-clock time) to gain any practical usage. In this thesis, we explore two general solutions to tackle this problem. Our first proposed method is a multiscaling approach. Simulating physics at its fundamental discrete scale, e.g., the atomic-level, provides unmatched levels of detail and generality, but proves to be excessively costly when applied to large-scale systems. Alternatively, simulating physics at the continuum scale governed by partial differential equations (PDEs) is computationally tractable, but limited in applicability due to built-in modeling assumptions. We propose a multiscaling simulation technique that exploits the dual strengths of discrete and continuum treatments.

In particular, we design a hybrid discrete-continuum framework for granular media. In this adaptive framework, we define an oracle to dynamically partition the domain into continuum regions where safe and discrete regions where necessary. We couple the dynamics of the discrete and continuum regions via overlapping transition zones to form one coherent simulation. Enrichment and homogenization operations convert between discrete and continuum representations, which allow the partitions to evolve over time. This approach saves the computation cost by partially employing continuum simulations and obtains up to 116X speedup over the discrete-only simulations while maintaining the same level of accuracy. To further accelerate PDE-governed continuum simulations, we propose a machine-learning-based reduced-order modeling (ROM) method. Whereas prior ROM approaches reduce the dimensionality of discretized vector fields, our continuous reduced-order modeling (CROM) approach builds a smooth, low-dimensional manifold of the continuous vector fields themselves, not their discretization.

We represent this reduced manifold using neural fields, relying on their continuous and differentiable nature to efficiently solve the PDEs. CROM may train on any and all available numerical solutions of the continuous system, even when they are obtained using diverse methods or discretizations. Indeed, CROM is the first model reduction framework that can simultaneously handle data from voxels, meshes, and point clouds. After the low-dimensional manifolds are established, solving PDEs requires significantly less computational resources. Since CROM is discretization-agnostic, CROM-based PDE solvers may optimally adapt discretization resolution over time to economize computation. We validate our approach on an extensive range of PDEs from thermodynamics, image processing, solid mechanics, and fluid dynamics. Selected large-scale experiments demonstrate that our approach obtains speed, memory, and accuracy advantages over prior ROM approaches while gaining 109X wall-clock speedup over full-order models on CPUs and 89X speedup on GPUs.