UPT++: Latent Point Set Neural Operators for Modeling System State Transitions

Particle methods comprise a wide spectrum of numerical algorithms, ranging from computational fluid dynamics governed by the Navier-Stokes equations to molecular dynamics governed by the many-body Schrödinger equation. At its core, these methods represent the continuum as a collection of discrete particles, on which the respective PDE is solved. We introduce UPT++, a latent point set neural operator for modeling the dynamics of such particle systems by mapping a particle set back to a continuous (latent) representation, instead of operating on the particles directly. We argue via what we call the discretization paradox that continuous modeling is advantageous even if the reference numerical discretization scheme comprises particles. Algorithmically, UPT++ extends Universal Physics Transformers -- a framework for efficiently scaling neural operators -- by novel importance-based encoding and decoding. Furthermore, our encoding and decoding enable outputs that remain consistent across varying input sampling resolutions, i.e., UPT++ is a neural operator. We discuss two types of UPT++ operators: (i) time-evolution operator for fluid dynamics, and (ii) sampling operator for molecular dynamics tasks. Experimentally, we demonstrate that our method reliably models complex physics phenomena of fluid dynamics and exhibits beneficial scaling properties, tested on simulations of up to 200k particles. Furthermore, we showcase on molecular dynamics simulations that UPT++ can effectively explore the metastable conformation states of unseen peptide molecules.