Micro-Macro Tensor Neural Surrogates for Uncertainty Quantification in Collisional Plasma

TL;DR

This work tackles uncertainty quantification for the high-dimensional Vlasov–Poisson–Landau system by integrating a high-fidelity, asymptotic-preserving solver with low-fidelity surrogates from VPFP and EP through a variance-reduced Monte Carlo framework. A novel tensor SPINN surrogate exploit a micro–macro decomposition, learning an anisotropic Maxwellian background and a velocity-space residual to efficiently approximate the distribution while maintaining asymptotic consistency. Calibration and windowed training of the VPFP surrogate, along with optimally weighted control variates, yield substantial variance reductions (often ~two orders of magnitude) and large computational savings, enabling tens of thousands of realizations at negligible marginal cost. The approach demonstrates robust performance across linear/nonlinear Landau damping, two-bubbles relaxation, and two-stream instability, with transferable gains across Knudsen numbers and stochastic dimensions, and provides a practical pathway for reliable UQ in complex kinetic plasma models.

Abstract

Plasma kinetic equations exhibit pronounced sensitivity to microscopic perturbations in model parameters and data, making reliable and efficient uncertainty quantification (UQ) essential for predictive simulations. However, the cost of uncertainty sampling, the high-dimensional phase space, and multiscale stiffness pose severe challenges to both computational efficiency and error control in traditional numerical methods. These aspects are further emphasized in presence of collisions where the high-dimensional nonlocal collision integrations and conservation properties pose severe constraints. To overcome this, we present a variance-reduced Monte Carlo framework for UQ in the Vlasov--Poisson--Landau (VPL) system, in which neural network surrogates replace the multiple costly evaluations of the Landau collision term. The method couples a high-fidelity, asymptotic-preserving VPL solver with inexpensive, strongly correlated surrogates based on the Vlasov--Poisson--Fokker--Planck (VPFP) and Euler--Poisson (EP) equations. For the surrogate models, we introduce a generalization of the separable physics-informed neural network (SPINN), developing a class of tensor neural networks based on an anisotropic micro-macro decomposition, to reduce velocity-moment costs, model complexity, and the curse of dimensionality. To further increase correlation with VPL, we calibrate the VPFP model and design an asymptotic-preserving SPINN whose small- and large-Knudsen limits recover the EP and VP systems, respectively. Numerical experiments show substantial variance reduction over standard Monte Carlo, accurate statistics with far fewer high-fidelity samples, and lower wall-clock time, while maintaining robustness to stochastic dimension.

Figures (14)

• Figure 1: The Tensor Neural Network.
• Figure 1: Left: the $L_1$ and $L_{\infty}$ errors between Landau operator $Q(f,f)$ and the calibrated Fokker--Planck operator $\mu P(f)$; Right: the model error with respect to VPL for standard model, calibrated model without data, and calibrated model augmented with VPL data.
• Figure 2: UQ-SPINN for Vlasov--Poisson--Fokker--Planck equation.
• Figure 2: Different decomposition. Left: background term $\widetilde{\mathcal{M}}$; Right: perturbation term $g$; Top: anisotropic reference Maxwellian; Middle: isotropic reference Maxwellian; Bottom: moment-matched Maxwellian.
• Figure 3: Training loss for Example \ref{['exam0']}.

Theorems & Definitions (6)

• Lemma 4.1: Approximation on $\Omega\times\mathbb{R}^{d_v}$
• Lemma 5.1
• Proof 1
• Remark 5.2
• Remark 6.1: Surrogate models and computational cost
• Remark 6.2: Surrogate error vs PINN error