Robert Joseph George

Robert Joseph George, Jiawei Zhao, Jean Kossaifi et al.

Fourier Neural Operators (FNO) offer a principled approach to solving challenging partial differential equations (PDE) such as turbulent flows. At the core of FNO is a spectral layer that leverages a discretization-convergent representation in the Fourier domain, and learns weights over a fixed set of frequencies. However, training FNO presents two significant challenges, particularly in large-scale, high-resolution applications: (i) Computing Fourier transform on high-resolution inputs is computationally intensive but necessary since fine-scale details are needed for solving many PDEs, such as fluid flows, (ii) selecting the relevant set of frequencies in the spectral layers is challenging, and too many modes can lead to overfitting, while too few can lead to underfitting. To address these issues, we introduce the Incremental Fourier Neural Operator (iFNO), which progressively increases both the number of frequency modes used by the model as well as the resolution of the training data. We empirically show that iFNO reduces total training time while maintaining or improving generalization performance across various datasets. Our method demonstrates a 10% lower testing error, using 20% fewer frequency modes compared to the existing Fourier Neural Operator, while also achieving a 30% faster training.

Jean Kossaifi, Nikola Kovachki, Zongyi Li et al.

We present NeuralOperator, an open-source Python library for operator learning. Neural operators generalize neural networks to maps between function spaces instead of finite-dimensional Euclidean spaces. They can be trained and inferenced on input and output functions given at various discretizations, satisfying a discretization convergence properties. Built on top of PyTorch, NeuralOperator provides all the tools for training and deploying neural operator models, as well as developing new ones, in a high-quality, tested, open-source package. It combines cutting-edge models and customizability with a gentle learning curve and simple user interface for newcomers.

Suozhi Huang, Peiyang Song, Robert Joseph George et al.

Mathematical reasoning remains a significant challenge for Large Language Models (LLMs) due to hallucinations. When combined with formal proof assistants like Lean, these hallucinations can be eliminated through rigorous verification, making theorem proving reliable. However, even with formal verification, LLMs still struggle with long proofs and complex mathematical formalizations. While Lean with LLMs offers valuable assistance with retrieving lemmas, generating tactics, or even complete proofs, it lacks a crucial capability: providing a sense of proof progress. This limitation particularly impacts the overall development efficiency in large formalization projects. We introduce LeanProgress, a method that predicts the progress in the proof. Training and evaluating our models made on a large corpus of Lean proofs from Lean Workbook Plus and Mathlib4 and how many steps remain to complete it, we employ data preprocessing and balancing techniques to handle the skewed distribution of proof lengths. Our experiments show that LeanProgress achieves an overall prediction accuracy of 75.1\% in predicting the amount of progress and, hence, the remaining number of steps. When integrated into a best-first search framework using Reprover, our method shows a 3.8\% improvement on Mathlib4 compared to baseline performances of 41.2\%, particularly for longer proofs. These results demonstrate how proof progress prediction can enhance both automated and interactive theorem proving, enabling users to make more informed decisions about proof strategies.

Sebastian Loeschcke, David Pitt, Robert Joseph George et al.

Scientific problems require resolving multi-scale phenomena across different resolutions and learning solution operators in infinite-dimensional function spaces. Neural operators provide a powerful framework for this, using tensor-parameterized layers to capture complex, multi-dimensional relationships. However, scaling neural operators to high-resolution problems leads to significant computational demands, making the training of industrial-scale models prohibitive. In this work, we introduce \textbf{TensorGRaD}, a novel method that directly addresses the memory challenges associated with optimizing large tensor-structured weights. Our approach, based on a \texit{robust tensor decomposition}, factorizes gradients as the sum of a low-rank tensor and a sparse one to efficiently capture information within optimizer states, including outliers. Additionally, we provide a recipe for mixed precision training of TensorGRaD, achieving further memory savings without sacrificing accuracy. We showcase the effectiveness of TensorGRaD on Fourier Neural Operators, a class of models crucial for solving partial differential equations (PDE). We provide theoretical guarantees for TensorGRaD, demonstrating its fundamental advantage over matrix-based gradient compression methods. We empirically demonstrate large improvements across various PDE tasks, including the challenging turbulent Navier-Stokes case at a Reynolds number of $10^5$. TensorGRaD reduces total memory usage by over $50\%$ while maintaining and sometimes even improving accuracy.

Robert Joseph George, Carson Eisenach, Udaya Ghai et al.

Large language models (LLMs) are good at generating code, but remain brittle for formal verification in systems like Lean4. A core scalability challenge is that verified synthesis requires consistent outputs across multiple artifacts: executable code, precise specifications, theorem statements, and ultimately proofs. Existing approaches rarely treat these as a unified pipeline. We present BRIDGE, a structured prompting framework that decomposes verification into three interconnected domains: Code (implementations), Specifications (formal intent), and Theorem Statements (constructive correctness claims), and elicits domain-specific intermediate reasoning to connect them. In Lean4, BRIDGE often adopts a code-first workflow, using the generated implementation as a semantic anchor for downstream specification and theorem statement generation. Across 178 algorithmic problems and five LLMs, BRIDGE improves Lean executable correctness by nearly 1.5x (pass at 5) over direct baselines and can be 2x more sample-efficient at inference time, requiring fewer samples per verified solution at comparable generation lengths. We further find that specification-driven prompting improves Python pass rates by up to 17.5 percent. Beyond inference-time prompting, supervised fine-tuning on BRIDGE-style reasoning traces yields nearly 1.5x higher Lean pass success than code-only SFT, indicating that these intermediate representations are learnable. BRIDGE provides a practical foundation for scaling verified synthesis and motivates future work on expert iteration and full proof generation.

Md Ashiqur Rahman, Robert Joseph George, Mogab Elleithy et al.

Existing neural operator architectures face challenges when solving multiphysics problems with coupled partial differential equations (PDEs) due to complex geometries, interactions between physical variables, and the limited amounts of high-resolution training data. To address these issues, we propose Codomain Attention Neural Operator (CoDA-NO), which tokenizes functions along the codomain or channel space, enabling self-supervised learning or pretraining of multiple PDE systems. Specifically, we extend positional encoding, self-attention, and normalization layers to function spaces. CoDA-NO can learn representations of different PDE systems with a single model. We evaluate CoDA-NO's potential as a backbone for learning multiphysics PDEs over multiple systems by considering few-shot learning settings. On complex downstream tasks with limited data, such as fluid flow simulations, fluid-structure interactions, and Rayleigh-Bénard convection, we found CoDA-NO to outperform existing methods by over 36%.