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Weak adversarial tensor neural network for high-dimensional Fokker-Planck equations

Sihong Shao, Zhongwei Xu

arXiv:2610.32169v1math.NAcs.NA

TL;DR

High-dimensional stationary Fokker–Planck equations require normalization and costly high-dimensional integral computation. WATNN combines a weak adversarial formulation with tensor neural networks and fixed quadrature. It reports mean relative errors from 1.13% to 7.93% across four tested problems from 6D to 20D.

  • Problem

    High-dimensional stationary Fokker–Planck methods face normalization and costly high-dimensional integrals, while deep-learning residual minimization also admits the trivial zero solution without normalization.

    Source: §1 · p. 1

  • Method

    WATNN combines a weak adversarial formulation with tensor neural networks for the primal and test networks, using fixed quadrature for loss computation and normalization.

    Source: §1 · p. 2, §5 · p. 19

  • Results

    Mean relative errors were 1.50%, 1.13%, 1.20%, and 7.93% for the tested 6D unimodal, 6D multi-modal, 10D multi-modal, and 20D multi-modal problems, respectively.

    Source: §1 · p. 3, §5 · p. 19

  • Takeaways & Limitations

    Numerical experiments on 6D–20D problems show WATNN’s reported advantages over existing methods in accuracy and efficiency.

    Source: §5 · p. 19

    Applicability is limited when equation coefficients or nonlinear terms lack suitable tensor-product representations; domain estimation also relies on SDE simulations.

    Source: §5 · p. 19

Abstract

from arXiv · show

Solving high-dimensional stationary Fokker-Planck (SFP) equations remains challenging. Recently, deep learning methods have shown great potential for solving high-dimensional SFP equations due to their universal approximation and mesh-free training. However, the high-dimensional integrals involved in loss computation and normalization can be expensive to approximate accurately using Monte Carlo integration. To address this issue, we propose a weak adversarial tensor neural network (WATNN) method for solving high-dimensional SFP equations, which combines a weak adversarial (WA) formulation with the tensor neural network (TNN). The WA formulation avoids the cross terms introduced by squaring the strong residual, reducing the number of integrals to be computed. When both the primal network and the adversarial test network are set as TNNs, the TNN quadrature scheme provides accurate and efficient normalization and loss computation with relatively few fixed quadrature points, avoiding the statistical sampling error that arises when using Monte Carlo integration. We demonstrate the effectiveness numerically on four high-dimensional SFP equations. For the 6D unimodal, 6D multi-modal, 10D multi-modal and 20D multi-modal problems, WATNN achieves the mean relative errors of 1.50%, 1.13%, 1.20% and 7.93% respectively, in only a few minutes.

1. Introduction.

High-dimensional stationary Fokker–Planck equations challenge numerical methods and deep-learning approaches because of normalization and costly high-dimensional integrals. WATNN combines a weak adversarial formulation with tensor neural networks, and reports results on four problems from 6D to 20D.

1. Introduction.

  • Grid methods face dimensionality-driven growth in degrees of freedom, while Monte Carlo integration can require many samples and still leaves full-density reconstruction challenging.Deep-learning methods also face the homogeneous stationary equation’s trivial zero solution unless normalization is enforced.Source: §1 · p. 1
  • Prior SFP methods address normalization through density constraints, simulation data, or penalty terms [1]; score-based learning and weak sampling provide alternatives.Source: §1 · p. 2
  • WATNN combines a weak adversarial formulation with tensor neural networks for both primal and test networks, using fixed-node quadrature for the loss and normalization.Its weak residual avoids the cross terms introduced by squaring the strong residual.Source: §1 · pp. 2–3
  • TNN structure reduces tensor-product integrals to products of one-dimensional integrals, enabling fixed quadrature instead of Monte Carlo sampling.The paper motivates this approach because squaring the strong residual introduces cross terms and additional integrals.Source: §1 · p. 2
  • The method optimizes an adversarial test space spanned by rank-one components and trains directly from fixed quadrature nodes without invariant-distribution samples.SDE simulations are used only to estimate the computational domain.Source: §1 · p. 2
  • Mean relative errors were 1.50%, 1.13%, 1.20%, and 7.93% on the 6D unimodal, 6D multi-modal, 10D multi-modal, and 20D multi-modal problems, respectively.The reported runtimes were 131.28 s, 121.50 s, 166.94 s, and 170.87 s, respectively; the paper reports accuracy and efficiency advantages over FPNN on the tested examples.Source: §1 · p. 3

2. Preliminaries.

The preliminaries introduce the stationary Fokker–Planck setting on a bounded computational domain, then review weak adversarial networks and tensor neural networks. The TNN structure supports high-dimensional integration through products of one-dimensional calculations.

2. Preliminaries.

  • The SDE has drift and diffusion coefficients, and its invariant density satisfies the stationary Fokker–Planck equation; computations use a bounded domain intended to capture almost all probability mass.The density is assumed to decay rapidly near the domain boundary and vanish outside it.Source: §2 · p. 3

2.1. Weak adversarial network.

  • The weak adversarial method tests a weak residual against a neural test function and minimizes its operator norm, avoiding direct minimization of the strong residual.The strong residual requires second-order network derivatives, which can be computationally expensive in high dimensions.Source: §2.1 · pp. 3–4

2.2. Tensor neural network and its quadrature scheme.

  • A tensor neural network represents a multivariate function through products of one-dimensional subnetworks, allowing integrals of tensor-product expressions to decompose into one-dimensional integrals.This structure reduces high-dimensional integration to sums of products that can be approximated with standard one-dimensional quadrature.Source: §2.2 · p. 5
  • The computational domain is estimated from SDE simulations by selecting coordinate intervals whose product contains at least a 1−δ fraction of terminal samples.The described WATNN workflow then constructs the primal network and test space and computes its loss using TNN quadrature.Source: §2.2 · p. 6

3. Weak adversarial tensor neural network.

WATNN combines a weak adversarial formulation with tensor neural networks to approximate invariant densities on a bounded domain. Its weak residual and tensor-product structure support quadrature-based loss computation and alternating optimization of the density network and test space.

3.1. Computational domain and weak adversarial formulation.

  • On a bounded computational domain, WATNN seeks a density whose weak residual vanishes for every test function, omitting boundary terms under a density-decay assumption.The domain is estimated from SDE trajectories before training.Source: §3.1 · pp. 6–7

3.3. Test space and the loss function.

  • A finite-dimensional test space spanned by rank-one components defines the adversary, whose optimal test choice maximizes the loss while the primal network minimizes it.The test components are normalized in L2, preserving their span.Source: §3.1 · p. 7, §3.3 · p. 8

3.2. Normalized primal TNN.

  • The primal network is a nonnegative, normalized tensor construction assembled from coordinate-wise factors that are normalized over their respective intervals.The factors use softplus outputs; normalization constants are approximated with composite Gauss-Legendre quadrature.Source: §3.2 · pp. 7–8

3.4. Computation of the loss function using the TNN quadrature scheme.

  • TNN quadrature computes the residual vector and H1 Gram matrix through one-dimensional integrals, using only first-order spatial derivatives.The weak-residual loss uses d + d^2 integrals, compared with d^2 + d^3 + d^4 for the strong-residual loss.Source: §3.4 · pp. 9–10
  • Under the density-decay assumption, boundary integrals are negligible and test functions need not vanish on ∂Ω; the weak residual uses d + d^2 integrals and only first-order spatial network derivatives.Source: §3.1 · p. 7, §3.4 · p. 10

3.5. Training algorithm.

  • Training alternates Kη test-space maximization steps with Kθ primal-network minimization steps, then returns pθ as the invariant-density approximation.The computational domain is estimated from SDE trajectories before training, and fixed quadrature nodes are generated on each coordinate interval.Source: §3.5 · pp. 10–11

4. Numerical Experiments.

The experiments test WATNN on four SFP benchmarks spanning 6D to 20D, with known invariant densities and varied multimodal structures. Across these cases, WATNN reports low mean relative errors and favorable comparisons with FPNN.

4.1. Experiment setup.

  • The four benchmark problems use known invariant densities; computational domains and quadrature settings are selected for the experiments.Each coordinate interval is divided into M subintervals, with 16 Gauss-Legendre points per subinterval.Source: §4.1 · pp. 11–12, Table 4.1 · p. 13

4.2. 6D Unimodal problem.

4.3. 6D Multi-modal problem.

4.4. 10D Multi-modal problem.

4.5. 20D Multi-modal problem.

  • 7.93% MRE in 170.87 s after 20k steps fell to 4.31% after 80k steps (610.88 s); TNN-FPNN returned NaN and MLP-FPNN errors were 95.88% and 73.02%.WATNN's projections also capture the multimodal structure and low-density regions, including density around 10^-4.Source: Table 4.8 · p. 18, §4.5 · p. 18

5. Conclusion.

WATNN combines the weak adversarial formulation with tensor neural networks for both primal and test networks, using a TNN quadrature scheme for normalization and loss computation. Experiments on 6–20D SFP problems demonstrate its potential, while its applicability is limited when equation terms lack suitable tensor-product representations.

5. Conclusion.

  • WATNN applies the weak adversarial formulation to SFP equations, represents both primal and test networks with TNNs, and uses TNN quadrature to compute loss and enforce normalization.The method optimizes a test space spanned by rank-one components of the test TNN and uses relatively few fixed quadrature points.Source: §5 · pp. 18–19
  • Numerical experiments on challenging 6–20D SFP problems demonstrate WATNN's potential and advantages over existing methods in accuracy and efficiency.For the 20D multi-modal problem, Fig. 4.4 shows contour projections of the exact and WATNN solutions, with remaining coordinates fixed at zero; Table 4.9 reports FPNN results for TNN/MLP architectures.Source: §5 · p. 19, Figure 4.4 · p. 19, Table 4.9 · p. 19
  • When equation coefficients or nonlinear terms lack suitable tensor-product representations, TNN quadrature cannot be directly applied, limiting WATNN's applicability to more general equations.The current computational-domain estimation also relies on SDE simulations, which may restrict use on unbounded domains without a probabilistic interpretation.Source: §5 · p. 19
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