Vasily Ilin

Tyson Klingner, Drew Bladek, Escher Crawford et al.

Within the past few years, the ability of Large Language Models (LLMs) to generate formal mathematical proofs has improved drastically. We provide a comparison of various LLMs' effectiveness in producing formal proofs in Lean 4 with the goal of assisting those seeking to use LLMs to support their own projects. We utilize both pass@$k$ and refine@$k$ metrics as the benchmark for our comparison and evaluate on subsets of both miniF2F and miniCTX datasets. Our testing shows that overall, Gemini 3.1 Pro and Claude Opus 4.7 perform best. Gemini 3.1 Pro achieved a 92\% success rate on miniF2F via refine@32 whereas Opus 4.7 achieved a 86\% success rate on miniCTX via refine@32. When taking cost into account, NVIDIA Nemotron 3 Super and GPT-OSS 120B were the most efficient, with competitive accuracies and average costs of $<\$0.01$ per correct proof.

Luke Alexander, Eric Leonen, Sophie Szeto et al.

Searching for mathematical results remains difficult: most existing tools retrieve entire papers, while mathematicians and theorem-proving agents often seek a specific theorem, lemma, or proposition that answers a query. While semantic search has seen rapid progress, its behavior on large, highly technical corpora such as research-level mathematical theorems remains poorly understood. In this work, we introduce and study semantic theorem retrieval at scale over a unified corpus of $9.2$ million theorem statements extracted from arXiv and seven other sources, representing the largest publicly available corpus of human-authored, research-level theorems. We represent each theorem with a short natural-language description as a retrieval representation and systematically analyze how representation context, language model choice, embedding model, and prompting strategy affect retrieval quality. On a curated evaluation set of theorem-search queries written by professional mathematicians, our approach substantially improves both theorem-level and paper-level retrieval compared to existing baselines, demonstrating that semantic theorem search is feasible and effective at web scale. The theorem search tool is available at \href{https://huggingface.co/spaces/uw-math-ai/theorem-search}{this link}, and the dataset is available at \href{https://huggingface.co/datasets/uw-math-ai/TheoremSearch}{this link}.

Evan Wang, Simon Chess, Daniel Lee et al.

As neural theorem provers become increasingly agentic, the ability to interpret and act on compiler feedback is critical. However, existing Lean datasets consist almost exclusively of correct proofs, offering little supervision for understanding and repairing failures. We study Lean proof repair as a supervised learning problem: given an erroneous proof and compiler feedback, predict both a corrected proof and a natural-language diagnosis grounded in the same feedback. We introduce APRIL (Automated Proof Repair in Lean), a dataset of 260,000 supervised tuples pairing systematically generated proof failures with compiler diagnostics and aligned repair and explanation targets. Training language models on APRIL substantially improves repair accuracy and feedback-conditioned reasoning; in our single-shot repair evaluation setting, a finetuned 4B-parameter model outperforms the strongest open-source baseline. We view diagnostic-conditioned supervision as a complementary training signal for feedback-using provers. Our dataset is available at \href{https://huggingface.co/datasets/uw-math-ai/APRIL}{this link}.

Vasily Ilin

We present a complete Lean 4 formalization of the equilibrium characterization in the Vlasov-Maxwell-Landau (VML) system, which describes the motion of charged plasma. The project demonstrates the full AI-assisted mathematical research loop: an AI reasoning model (Gemini DeepThink) generated the proof from a conjecture, an agentic coding tool (Claude Code) translated it into Lean from natural-language prompts, a specialized prover (Aristotle) closed 111 lemmas, and the Lean kernel verified the result. A single mathematician supervised the process over 10 days at a cost of \$200, writing zero lines of code. The entire development process is public: all 229 human prompts, and 213 git commits are archived in the repository. We report detailed lessons on AI failure modes -- hypothesis creep, definition-alignment bugs, agent avoidance behaviors -- and on what worked: the abstract/concrete proof split, adversarial self-review, and the critical role of human review of key definitions and theorem statements. Notably, the formalization was completed before the final draft of the corresponding math paper was finished.

Vasily Ilin, Jingwei Hu

Plasma modeling is central to the design of nuclear fusion reactors, yet simulating collisional plasma kinetics from first principles remains a formidable computational challenge: the Vlasov-Maxwell-Landau (VML) system describes six-dimensional phase-space transport under self-consistent electromagnetic fields together with the nonlinear, nonlocal Landau collision operator. A recent deterministic particle method for the full VML system estimates the velocity score function via the blob method, a kernel-based approximation with $O(n^2)$ cost. In this work, we replace the blob score estimator with score-based transport modeling (SBTM), in which a neural network is trained on-the-fly via implicit score matching at $O(n)$ cost. We prove that the approximated collision operator preserves momentum and kinetic energy, and dissipates an estimated entropy. We also characterize the unique global steady state of the VML system and its electrostatic reduction, providing the ground truth for numerical validation. On three canonical benchmarks -- Landau damping, two-stream instability, and Weibel instability -- SBTM is more accurate than the blob method, achieves correct long-time relaxation to Maxwellian equilibrium where the blob method fails, and delivers $50\%$ faster runtime with $4\times$ lower peak memory.

Vasily Ilin, Peter Sushko

We introduce a unified attention-based framework for joint score and density estimation. Framing the problem as a sequence-to-sequence task, we develop a permutation- and affine-equivariant transformer that estimates both the probability density $f(x)$ and its score $\nabla_x \log f(x)$ directly from i.i.d. samples. Unlike traditional score-matching methods that require training a separate model for each distribution, our approach learns a single distribution-agnostic operator that generalizes across densities and sample sizes. The architecture employs cross-attention to connect observed samples with arbitrary query points, enabling generalization beyond the training data, while built-in symmetry constraints ensure equivariance to permutation and affine transformations. Analytically, we show that the attention weights can recover classical kernel density estimation (KDE), and verify it empirically, establishing a principled link between classical KDE and the transformer architecture. Empirically, the model achieves substantially lower error and better scaling than KDE and score-debiased KDE (SD-KDE), while exhibiting better runtime scaling. Together, these results establish transformers as general-purpose, data-adaptive operators for nonparametric density and score estimation.

Vasily Ilin, Jingwei Hu, Zhenfu Wang · pku

We propose a particle method for numerically solving the Landau equation, inspired by the score-based transport modeling (SBTM) method for the Fokker-Planck equation. This method can preserve some important physical properties of the Landau equation, such as the conservation of mass, momentum, and energy, and decay of estimated entropy. We prove that matching the gradient of the logarithm of the approximate solution is enough to recover the true solution to the Landau equation with Maxwellian molecules. Several numerical experiments in low and moderately high dimensions are performed, with particular emphasis on comparing the proposed method with the traditional particle or blob method.

Peter Sushko, Ayana Bharadwaj, Zhi Yang Lim et al. · allen-ai, uw

Existing image editing models struggle to meet real-world demands. Despite excelling in academic benchmarks, they have yet to be widely adopted for real user needs. Datasets that power these models use artificial edits, lacking the scale and ecological validity necessary to address the true diversity of user requests. We introduce REALEDIT, a large-scale image editing dataset with authentic user requests and human-made edits sourced from Reddit. REALEDIT includes a test set of 9300 examples to evaluate models on real user requests. Our results show that existing models fall short on these tasks, highlighting the need for realistic training data. To address this, we introduce 48K training examples and train our REALEDIT model, achieving substantial gains - outperforming competitors by up to 165 Elo points in human judgment and 92 percent relative improvement on the automated VIEScore metric. We deploy our model on Reddit, testing it on new requests, and receive positive feedback. Beyond image editing, we explore REALEDIT's potential in detecting edited images by partnering with a deepfake detection non-profit. Finetuning their model on REALEDIT data improves its F1-score by 14 percentage points, underscoring the dataset's value for broad applications.

Vasily Ilin, Peter Sushko, Jingwei Hu · allen-ai, uw

We propose a deterministic sampling framework using Score-Based Transport Modeling for sampling an unnormalized target density $π$ given only its score $\nabla \log π$. Our method approximates the Wasserstein gradient flow on $\mathrm{KL}(f_t\|π)$ by learning the time-varying score $\nabla \log f_t$ on the fly using score matching. While having the same marginal distribution as Langevin dynamics, our method produces smooth deterministic trajectories, resulting in monotone noise-free convergence. We prove that our method dissipates relative entropy at the same rate as the exact gradient flow, provided sufficient training. Numerical experiments validate our theoretical findings: our method converges at the optimal rate, has smooth trajectories, and is often more sample efficient than its stochastic counterpart. Experiments on high-dimensional image data show that our method produces high-quality generations in as few as 15 steps and exhibits natural exploratory behavior. The memory and runtime scale linearly in the sample size.