Neural Methods for Plasma Simulation and Mathematical Formalization

Abstract

This dissertation consists of several applications of neural networks to plasma simulation and to mathematical formalization. It synthesizes eight papers into seven thematic chapters across three parts. While nuclear fusion promises an abundance of clean energy, achieving it requires fast and accurate numerical methods to simulate plasma, whose collisional dynamics are governed by the Landau and Vlasov--Maxwell--Landau (VML) equations. I present a family of deterministic particle methods that exploit the fact that these equations can be written as continuity equations whose velocity field depends on the density only through the score function $\nabla \log f$. Replacing the kernel density estimate used in the classical blob method with a neural network trained online via implicit score matching produces a particle scheme that scales linearly rather than quadratically in the number of particles, conserves mass, momentum, and energy by construction, and avoids the unphysical truncation of velocity tails to which kernel methods are prone. The resulting solver is both faster and more accurate than the previous state of the art. The numerical advances are accompanied by theoretical guarantees: stability of the spatially homogeneous Coulomb-case Landau equation in relative entropy, an error bound on the score-based approximation in terms of the score-matching loss, and a characterization of the global equilibria of the VML and Vlasov--Poisson--Landau systems on the torus. The same score-based viewpoint extends beyond plasma to general distributions. I apply online score learning to a strictly harder problem than diffusion generative modeling -- deterministic sampling from an unnormalized target density given only its score and no samples -- and prove the optimal exponential entropy dissipation rate through a neural tangent kernel analysis of the coupled particle--network gradient flow. I then turn score estimation itself into the object of study and develop DiScoFormer, a permutation- and affine-equivariant Transformer that amortizes density and score estimation across distributions, sample sizes, and dimensions, with the property that a single self-attention head exactly reproduces Gaussian kernel density estimation -- making precise the relationship between attention and classical nonparametric statistics. While neural network-based AI promises an abundance of cheap intelligence, it comes at the cost of hallucinations. On the other hand, formal languages like Lean can guarantee complete correctness but are tedious to program in. I present several applications of AI to formalizing mathematics in Lean, organized as the discovery, repair, and formalization workflow of working mathematicians. We built a semantic search engine over theorem statements extracted from arXiv and the Stacks Project, which outperforms LLM-based search. We constructed APRIL, a dataset of broken-then-repaired Lean~4 proofs paired with compiler diagnostics, on which a fine-tuned small language model matches the state-of-the-art model at single-shot proof repair. Finally, I present a Lean~4 formalization of an equilibrium theorem for the Vlasov--Maxwell--Landau system, proved informally in Part~I. The experiment illustrates how AI and formal verification become complementary: verification eliminates the hallucinations of AI, while AI makes formalization practical at research scale.