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Bingguang Chen

Publications and source records attributed to Bingguang Chen.

7 recordsLinked to original sources
arXiv2026

SPDEBench: An Extensive Benchmark for Learning Stochastic PDEs

Stochastic Partial Differential Equations (SPDEs) driven by random noise play a central role in modeling physical processes with rough spatio-temporal dynamics, such as turbulence flows, superconductors, and quantum dynamics. Although machine learning (ML)-based surrogate models have shown promise for efficiently approximating such dynamics, progress remains limited by the lack of a unified benchmark with controlled data generation and comprehensive evaluation. This gap is particularly significant for singular SPDEs, for which benchmark datasets are largely unavailable and reliable simulation requires numerically delicate schemes based on renormalization. Moreover, subtle differences in data-generation procedures, such as noise approximation, basis choice, and the inclusion of renormalization, can significantly affect the resulting datasets and, consequently, model evaluation. We introduce SPDEBench, the first unified benchmark for ML-based SPDE learning. SPDEBench provides ready-to-use datasets for physically and mathematically significant SPDEs on 1-3D domains with periodic or Dirichlet boundary condition. Both regular and singular SPDEs are taken into consideration. SPDEBench also incorporates representative ML baselines in operator learning, together with 7 evaluation metrics, including Sobolev and distributional metrics beyond the standard $L^2$-error. Supported by SPDEBench, we conduct systematic evaluations of model accuracy, robustness, and out-of-distribution generalization under controlled data variations. Our numerical results show that SPDE-aware architectures generally achieve stronger performance than generic operator-learning baselines. These findings establish SPDEBench as a reproducible and extensible resource, paving pathway for principled benchmarking and architecture design for stochastic spatio-temporal dynamics.

cs.LG↗
arXiv2025

Riemannian Neural Geodesic Interpolant

Stochastic interpolants are efficient generative models that bridge two arbitrary probability density functions in finite time, enabling flexible generation from the source to the target distribution or vice versa. These models are primarily developed in Euclidean space, and are therefore limited in their application to many distribution learning problems defined on Riemannian manifolds in real-world scenarios. In this work, we introduce the Riemannian Neural Geodesic Interpolant (RNGI) model, which interpolates between two probability densities on a Riemannian manifold along the stochastic geodesics, and then samples from one endpoint as the final state using the continuous flow originating from the other endpoint. We prove that the temporal marginal density of RNGI solves a transport equation on the Riemannian manifold. After training the model's the neural velocity and score fields, we propose the Embedding Stochastic Differential Equation (E-SDE) algorithm for stochastic sampling of RNGI. E-SDE significantly improves the sampling quality by reducing the accumulated error caused by the excessive intrinsic discretization of Riemannian Brownian motion in the classical Geodesic Random Walk (GRW) algorithm. We also provide theoretical bounds on the generative bias measured in terms of KL-divergence. Finally, we demonstrate the effectiveness of the proposed RNGI and E-SDE through experiments conducted on both collected and synthetic distributions on S2 and SO(3).

cs.LG↗
arXiv2022

Deep Random Vortex Method for Simulation and Inference of Navier-Stokes Equations

Navier-Stokes equations are significant partial differential equations that describe the motion of fluids such as liquids and air. Due to the importance of Navier-Stokes equations, the development on efficient numerical schemes is important for both science and engineer. Recently, with the development of AI techniques, several approaches have been designed to integrate deep neural networks in simulating and inferring the fluid dynamics governed by incompressible Navier-Stokes equations, which can accelerate the simulation or inferring process in a mesh-free and differentiable way. In this paper, we point out that the capability of existing deep Navier-Stokes informed methods is limited to handle non-smooth or fractional equations, which are two critical situations in reality. To this end, we propose the \emph{Deep Random Vortex Method} (DRVM), which combines the neural network with a random vortex dynamics system equivalent to the Navier-Stokes equation. Specifically, the random vortex dynamics motivates a Monte Carlo based loss function for training the neural network, which avoids the calculation of derivatives through auto-differentiation. Therefore, DRVM not only can efficiently solve Navier-Stokes equations involving rough path, non-differentiable initial conditions and fractional operators, but also inherits the mesh-free and differentiable benefits of the deep-learning-based solver. We conduct experiments on the Cauchy problem, parametric solver learning, and the inverse problem of both 2-d and 3-d incompressible Navier-Stokes equations. The proposed method achieves accurate results for simulation and inference of Navier-Stokes equations. Especially for the cases that include singular initial conditions, DRVM significantly outperforms existing PINN method.

physics.flu-dyn↗
arXiv2022

Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs

Stochastic partial differential equations (SPDEs) are significant tools for modeling dynamics in many areas including atmospheric sciences and physics. Neural Operators, generations of neural networks with capability of learning maps between infinite-dimensional spaces, are strong tools for solving parametric PDEs. However, they lack the ability to modeling SPDEs which usually have poor regularity due to the driving noise. As the theory of regularity structure has achieved great successes in analyzing SPDEs and provides the concept model feature vectors that well-approximate SPDEs' solutions, we propose the Neural Operator with Regularity Structure (NORS) which incorporates the feature vectors for modeling dynamics driven by SPDEs. We conduct experiments on various of SPDEs including the dynamic Phi41 model and the 2d stochastic Navier-Stokes equation, and the results demonstrate that the NORS is resolution-invariant, efficient, and achieves one order of magnitude lower error with a modest amount of data.

cs.LG↗
arXiv2019

On the small time asymptotics of the dynamical $Φ^4_1$ model

In this paper, we establish a small time large deviation principle (small time asymptotics) for the dynamical $Φ^4_1$ model, which not only involves study of the space-time white noise with intensity $\sqrt{\varepsilon}$, but also the investigation of the effect of the small (with $\varepsilon$) nonlinear drift.

math.PR↗