Session VI — Numerical Methods and Scientific Machine Learning

Tuesday, October 27 · 9:00–10:30 a.m.
Ballroom B
Chair: Siyuan (Simon) Xing

Simulating Dynamics of Large Ensembles of Acoustically Bound Particles

Hannah Love · UC Merced

We aim to efficiently simulate hundreds to thousands of mm-sized particles driven by forces from acoustic multiple scattering. Preliminary simulations and experiments show novel collective behavior, including stable clustering. We simulate these systems by (1) solving a multiple scattering problem, (2) computing binding forces, and (3) moving particles according to those forces. We use the Method of Fundamental Solutions (MFS), a mesh-free boundary method that produces a dense block system. As particle number increases, repeatedly solving this system becomes expensive. We use block Gauss-Seidel (BGS), initialized with the previous solution, which requires few iterations. Algorithmic modifications improve BGS efficiency while minimizing storage, enabling simulations with hundreds of particles on a single workstation. Results show stable collective structures dependent on particle size and density. We discuss extending this framework to larger systems and complex shapes and materials.

Teaching a new dog old tricks: Leveraging interpolation techniques for verifiable scientific ML

Andrew Gillette · Lawrence Livermore National Laboratory

Effective verification and validation techniques for modern scientific machine learning workflows are challenging to devise. We present an approach that combines error bounds for classical interpolation techniques with modern statistical methods and demonstrate that: (1) multiple standard interpolation techniques have informative error bounds that can be computed or estimated efficiently; (2) comparative performance among distinct interpolants can aid in validation goals; (3) deploying interpolation methods on latent spaces generated by deep learning techniques enables some interpretability for black-box models. We present a detailed case study of our approach for predicting lift-drag ratios from airfoil images. Code developed for this work is available in a public GitHub repository. This is joint work with Tyler Chang (formerly at Argonne National Laboratory) and Romit Maulik (Purdue University).

Compositional Physics-Informed Neural Flow

Pratham Lalwani · UC Merced

Scientific machine learning (SciML) is becoming a prominent alternative approach to solve differential equations over traditional numerical methods. While such approaches can enable fast inference of solutions with reasonable accuracy, SciML methods typically lack deterministic error bounds when extrapolated beyond their training domains. We propose Compositional Physics-Informed Neural Flow (CPINF) as an alternate SciML method to improve long-term accuracy of ODE/PDE solutions by composing their approximated flow maps over a short duration. For positively invariant CPINF, we establish rigorous a posteriori error estimates guaranteeing accuracy beyond its training time domain and identify analogous notions of A-stability and CFL condition. CPINF also has the practical advantage over other SciML methods of reducing memory cost and training complexity. We verify the theoretical results and benchmark CPINF versus both traditional and current SciML methods over a wide range of ODE/PDEs.

A symplectic mapping method with application to the Vlasov--Poisson equation

Seth Taylor · University of Saskatchewan

The Poisson structure of plasma models governs essential conservation laws, equilibria, and phase-space geometry. Numerical methods that preserve this structure exhibit improved long-term accuracy and physical consistency. In this talk, we present a geometric discretization for a family of Lie--Poisson equations based on a conforming discretization of the symplectomorphism group. By spatially discretizing the generating function representation of a canonical transformation, we obtain implicit phase-space maps that remain exactly symplectic. The resulting transported particle distributions conserve all Casimir invariants to machine precision. We will describe time integration schemes based on the solution of a Hamilton--Jacobi equation, together with efficient methods for evaluating the implicit symplectic maps. Numerical experiments for the 1+1 dimensional Vlasov--Poisson equation demonstrate high-order accuracy, Casimir conservation, and the resolution of fine-scale plasma dynamics.

Scalable Conservative Hamiltonian Monte Carlo

John P Gallagher · UC Merced

Markov chain Monte Carlo is a cornerstone method for sampling from target distributions, with applications to uncertainty quantification, Bayesian inference and generative models. Hamiltonian Monte Carlo (HMC) exploits Hamiltonian dynamics to efficiently draw samples using symplectic integrators such as leapfrog, but these can incur large energy errors for distributions with thin high-density regions, lowering acceptance probabilities for the generated samples. McGregor and Wan '26 proposed Conservative HMC (CHMC) using energy-preserving integrators to improve acceptance probability and convergence to the target distribution, at the cost of solving the resulting implicit schemes. We study scaling CHMC for high-dimensional applications using nonlinear solvers including fixed-point iteration, Anderson acceleration and Jacobian-free Newton-Krylov methods. Preliminary results show improved convergence of CHMC over HMC with competitive computational costs on various target distributions.