arXiv subjects

Andrew Qing He

Publications and source records attributed to Andrew Qing He.

8 recordsLinked to original sources

arXiv2026

Weak Adversarial Neural Pushforward Method for Boltzmann Equation

In this paper, we extend a weak adversary neural network pushforward method for solving time dependent Boltzmann equation and a weak formulation of the collision operator is proposed where an invertible neural pushforward mapping is used to generating samples given by the distribution governed by the Boltzmann equation. The training of the pushforward mapping is learnt by enforcing the weak form of the Boltzmann equation. Numerical results have demonstrated the effectiveness of the proposed method.

math.NA↗

arXiv2026

Weak Adversarial Neural Pushforward Method for the Wigner Transport Equation

We extend the Weak Adversarial Neural Pushforward Method to the Wigner transport equation governing the phase-space dynamics of quantum systems. The central contribution is a structural observation: integrating the nonlocal pseudo-differential potential operator against plane-wave test functions produces a Dirac delta that exactly inverts the Fourier transform defining the Wigner potential kernel, reducing the operator to a pointwise finite difference of the potential at two shifted arguments. This holds in arbitrary dimension, requires no truncation of the Moyal series, and treats the potential as a black-box function oracle with no derivative information. To handle the negativity of the Wigner quasi-probability distribution, we introduce a signed pushforward architecture that decomposes the solution into two non-negative phase-space distributions mixed with a learnable weight. The resulting method inherits the mesh-free, Jacobian-free, and scalable properties of the original framework while extending it to the quantum setting.

quant-ph↗

arXiv2026

Deep Kuratowski Embedding Neural Networks for Wasserstein Metric Learning

Computing pairwise Wasserstein distances is a fundamental bottleneck in data analysis pipelines. Motivated by the classical Kuratowski embedding theorem, we propose two neural architectures for learning to approximate the Wasserstein-2 distance ($W_2$) from data. The first, DeepKENN, aggregates distances across all intermediate feature maps of a CNN using learnable positive weights. The second, ODE-KENN, replaces the discrete layer stack with a Neural ODE, embedding each input into the infinite-dimensional Banach space $C^1([0,1], \mathbb{R}^d)$ and providing implicit regularization via trajectory smoothness. Experiments on MNIST with exact precomputed $W_2$ distances show that ODE-KENN achieves a 28% lower test MSE than the single-layer baseline and 18% lower than DeepKENN under matched parameter counts, while exhibiting a smaller generalization gap. The resulting fast surrogate can replace the expensive $W_2$ oracle in downstream pairwise distance computations.

cs.LG↗

arXiv2026

Weak Adversarial Neural Pushforward Method for the McKean-Vlasov / Mean-Field Fokker-Planck Equation

We extend the Weak Adversarial Neural Pushforward Method (WANPM) to the McKean--Vlasov mean-field Fokker--Planck equation, covering both the stationary and time-dependent cases. The key observation is that the mean-field nonlinearity -- an expectation under the solution distribution -- is naturally estimated by Monte Carlo sampling from the pushforward network, requiring no change to the architecture and only minor modifications to the training loop. For the quadratic (granular media) interaction kernel, the interaction term reduces to the batch sample mean, eliminating secondary sampling entirely. We also identify a dimension-dependent frequency initialization rule for the adversarial test functions, necessary to avoid spurious minimizers. Numerical experiments on linear McKean--Vlasov benchmarks in 2, 5, 20, and 100 dimensions confirm accurate recovery of the exact Gaussian stationary and transient distributions, with training times ranging from 27 seconds (2D) to 10 minutes (100D) on a single GPU.

math.NA↗

arXiv2026

Weak Adversarial Neural Pushforward Method for Fractional Fokker-Planck Equations

We extend the Weak Adversarial Neural Pushforward Method (WANPM) to fractional Fokker-Planck equations, in which the classical Laplacian diffusion operator is replaced by the fractional Laplacian of order alpha in (0, 2]. The solution distribution is represented as the pushforward of a simple base distribution through a neural network, and the weak formulation is discretized entirely via Monte Carlo sampling without any temporal mesh. A key computational advantage is that plane-wave test functions are eigenfunctions of the fractional Laplacian, making the operator cost identical to that of classical diffusion for any alpha. We validate the method on seven benchmark problems with alpha = 1.5, spanning one and two spatial dimensions: the steady-state fractional Ornstein--Uhlenbeck (OU) process, a harmonic confining potential, a double-well potential, and a triple-well potential in one dimension, a steady-state 2D double-peak distribution, a time-dependent 2D ring distribution with rotational drift, and a five-dimensional harmonic potential. Each case is benchmarked against particle simulations using symmetric alpha-stable Lévy increments, and robust statistics confirm close agreement throughout. The method is mesh-free, requires no density evaluation or non-local quadrature, and provides a promising foundation for high-dimensional anomalous diffusion solvers.

math.NA↗

arXiv2026

Neural Pushforward Samplers for the Fokker-Planck Equation on Embedded Riemannian Manifolds

In this paper, we extend the Weak Adversarial Neural Pushforward Method to the Fokker--Planck equation on compact embedded Riemannian manifolds. The method represents the solution as a probability distribution via a neural pushforward map that is constrained to the manifold by a retraction layer, enforcing manifold membership and probability conservation by construction. Training is guided by a weak adversarial objective using ambient plane-wave test functions, whose intrinsic differential operators are derived in closed form from the geometry of the embedding, yielding a fully mesh-free and chart-free algorithm. Both steady-state and time-dependent formulations are developed, and numerical results on a double-well problem on the two-sphere demonstrate the capability of the method in capturing multimodal invariant distributions on curved spaces.

math.NA↗

arXiv2025

Learning Neural Pushforward Samplers for Distributions from Fokker-Planck Equations by Weak Adversarial Training

This paper presents a new method for solving Fokker-Planck equations (FPE) by learning a neural sampler for the distribution given by the FPE via an adversarial training based on a weak formulation of the FPE where the adjoint operator of FPE acts on the test function. Such a weak formulation transforms the PDE solution problem into a Monte Carlo importance sampling problem where the FPE solution-distribution is learned through a neural pushforward map, avoiding some of the limitations of direct PDE based methods. Moreover, by using simple plane-wave test functions, derivatives on the test functions can be explicitly computed. This approach produces a natural importance sampling strategy for the FPE solution distribution with probability conservation, from which the FPE solution can be easily constructed.

math.NA↗

arXiv2025

ARDO: A Weak Formulation Deep Neural Network Method for Elliptic and Parabolic PDEs Based on Random Differences of Test Functions

We propose ARDO method for solving PDEs and PDE-related problems with deep learning techniques. This method uses a weak adversarial formulation but transfers the random difference operator onto the test function. The main advantage of this framework is that it is fully derivative-free with respect to the solution neural network. This framework is particularly suitable for Fokker-Planck type second-order elliptic and parabolic PDEs.

math.NA↗