The paper addresses the Maxey‑Riley‑Gatignol equations, which describe the motion of spherical inertial particles in a fluid and include the Basset force—a history‑dependent integral term. Because this term complicates numerical solutions, it is often omitted, yet it significantly influences particle trajectories. The authors propose using universal differential equations, specifically neural networks, to approximate the Basset force, thereby converting the equations into a system of ordinary differential equations solvable by standard methods such as Runge‑Kutta.
arXiv:2607. 02194v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers.
arXiv:2602.08515v3 Announce Type: replace-cross Abstract: This work investigates shallow physics-informed neural networks (PINNs) for solving forward and inverse problems governed by nonlinear partia...
arXiv:2606. 19754v1 Announce Type: new Abstract: Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems.
arXiv:2601. 00473v4 Announce Type: replace-cross Abstract: We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms.
arXiv:2609.33078v2 Announce Type: replace Abstract: Automatic differentiation (AD) lets neural networks compute derivatives of governing equations to machine precision, and this precision has made it...
arXiv:2607. 28762v1 Announce Type: new Abstract: This work embeds feature interaction modules derived from factorization machines (FMs) into physics-informed neural networks (PINNs) and neural operator learning, to enhance model expressiveness for solution manifolds of parameterized partial differential equations (PDEs).
arXiv:2608. 10389v1 Announce Type: cross Abstract: In recent years, neural networks have significantly advanced numerical solutions of partial differential equations (PDEs).
arXiv:2606. 18175v1 Announce Type: cross Abstract: We present a numerical method for the forward solution of nonlinear partial differential equations (PDEs) in which Bellman-Kalaba quasilinearization reduces the nonlinear problem to a sequence of linear subproblems, each discretized by collocation onto a trial space that is linear in its parameters and solved by a single direct linear least-squares QR factorization.
arXiv:2608.23022v1 Announce Type: cross Abstract: The dynamics of particles in the early universe are described by Boltzmann equations, which involve high-dimensional phase-space integrals. Classical...
DeepSPoC is a neural particle method that replaces direct particle-particle interactions in sequential propagation of chaos (SPoC) with particle‑network interactions, using a neural density representation (KRnet) to approximate the empirical measure. By simulating particles in batches and embedding a neural network into the mean‑field SDE coefficients, DeepSPoC reduces memory usage and computational cost compared to traditional particle methods. The approach is demonstrated on various mean‑field equations, showing improved scalability for high‑dimensional problems.
arXiv:2609.36615v1 Announce Type: cross Abstract: Physics-informed neural networks (PINNs) solve partial differential equations (PDEs) by incorporating governing physical laws into the training loss....
arXiv:2607. 15751v1 Announce Type: new Abstract: This work introduces Physics-Informed Splines (PI-Splines), a structured spline-based architecture for physics-informed learning.