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新论文 · v1arXiv:2611.14979

Coarse-to-Fine 3D MRI Reconstruction via Resolution-Agnostic Neural Operators

Jiayun Wang, Ram Daftari, Weijie Gan, Valentin Duruisseaux, Ruibo Wang, Armeet Singh Jatyani, Boris Bonev, Anima Anandkumar

原文摘要

Deep-learning-based 3D MRI reconstruction is limited in practice by the GPU memory and compute cost of full-volume training; 3D acquisitions are thus most often reconstructed by 2D models applied slice-by-slice and then stacked, which ignores through-plane (stacking-axis) structure. We reconstruct in full 3D and ask whether a model can be trained at low resolution (fast and memory-feasible) yet deployed at higher resolution. This coarse-to-fine strategy needs reliable cross-resolution performance, but standard networks are resolution-specific and degrade substantially when evaluated outside the training setting. We therefore use resolution-agnostic neural operators (NOs) and propose KIR-NO (k-space-to-Image Reconstruction Neural Operator), a discretization-agnostic framework for accelerated 3D MRI reconstruction. KIR-NO is trained with full backpropagation on low-resolution volumes and applied zero-shot to higher-resolution volumes at inference. Its core building block is a 3D neural operator based on discrete-continuous convolutions: convolutional filters are parameterized in a continuous basis and sampled at arbitrary voxel resolutions while preserving the local inductive bias of convolution. KIR-NO integrates these learned operators with physics-based consistency: a k-space neural-operator head first refines the undersampled Fourier measurements, and image-space neural-operator cascades then refine the reconstruction through interleaved data-consistency updates. Across 3D knee and brain MRI benchmarks, KIR-NO improves over classical and CNN baselines: it improves over a state-of-the-art 3D CNN by 2.2 dB PSNR at half resolution under 4x acceleration on SKM-TEA, with gains persisting at 8x and 16x. These results show that resolution-agnostic operators make coarse-to-fine 3D MRI reconstruction - cheap low-resolution training with high-resolution deployment - practical across anatomies.

eess.IV · cs.CV

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2609.02982

Equation Recast for Canonical Operator Learning Across Parametric PDEs

Qiyun Cheng, Valentin Duruisseaux, Cesar F. Clauser, Md Hossain Sahadath, Huihua Yang, Shaowu Pan, Nathaniel Ferraro, Anima Anandkumar 等 10 位作者

原文摘要

Learning solution operators across broad parameter ranges can require substantial coverage of both input functions and physical parameters, particularly for purely data-driven parametric models. In addition, the resulting models may fail silently outside the training distribution. We introduce equation recast, which reformulates parametric operator learning as the learning of a single canonical operator. Parameter-induced operator variations are derived analytically from the governing equation and absorbed into effective sources, enabling zero-shot prediction across new parameter regimes. Across multi-parameter, nonlinear, and singular PDE settings, equation recast supports extrapolation, integrates sparse heterogeneous datasets in a shared canonical representation, and uses loss of convergence as an internal warning signal for failure of the recast iteration. In high-fidelity tokamak simulations for nuclear fusion, the framework unifies electron-temperature data across four device geometries through canonical-domain mapping within one jointly trained operator. Equation recast provides a route toward reusable neural PDE solvers combining equation-guided transfer, data efficiency, and monitorable inference.

cs.LG · physics.comp-ph · physics.plasm-ph

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2609.36056

GeoWind2Plan: Mission-Time 3D Urban Wind Prediction for Energy-Efficient UAV Planning

Shaoxiang Qin, Yucheng Zhao, Fuyuan Lyu, Di Zhou, Jiachen Yao, Xue Liu, Anima Anandkumar, Liangzhu Leon Wang 等 9 位作者

原文摘要

In urban low-altitude flight, buildings reshape ambient wind into spatially varying 3D flow, making unmanned aerial vehicle (UAV) energy depend on local wind exposure as well as path length. However, building-resolved wind is rarely available when a mission must be planned. Computational fluid dynamics (CFD) produces high-fidelity urban flow fields, but each simulation is tied to a fixed inflow boundary condition and can take hours to days, which is incompatible with urban UAV missions that typically last minutes to tens of minutes. We present GeoWind2Plan, a geometry-to-wind-to-planning framework for mission-time 3D urban wind prediction and energy-efficient UAV planning. Given only a background wind vector, 3D building geometry, and a start-goal pair, GeoWind2Plan transforms the building geometry into a reference-wind frame, predicts mission-relevant 3D wind patches with a localized geometry-conditioned neural operator, stitches them into a local wind field, and optimizes a feasible 3D path and speed profile using a physically grounded UAV energy model. Rather than pursuing CFD-perfect reconstruction, GeoWind2Plan targets decision-useful wind prediction: trajectories are planned with predicted wind and evaluated under high-fidelity CFD wind. Across held-out urban domains, wind speeds, and mission wind-angle regimes, GeoWind2Plan performs corridor-localized wind inference in about 3 s, compared with roughly 8 h for CFD. Under CFD evaluation, trajectories planned with GeoWind2Plan reduce energy by 6.9%, 12.7%, and 4.5% in tailwind, headwind, and crosswind missions relative to wind-agnostic planning, recovering 87.9%, 85.7%, and 75.0% of CFD-reference savings. These results show that fast, corridor-localized 3D urban wind prediction can make wind-aware UAV energy planning practical at mission time. Code/dataset: https://github.com/DUAL-Xiao/GeoWind2Plan/ . Demo: https://geowind2plan.com .

cs.AI · cs.RO

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arXiv ↗PDF ↗
版本更新 · v5arXiv:2509.03738

Mechanistic Interpretability with Sparse Autoencoder Neural Operators

Bahareh Tolooshams, Ailsa Shen, Anima Anandkumar

原文摘要

We introduce sparse autoencoder neural operators (SAE-NOs), a new class of sparse autoencoders that operate in function spaces rather than fixed-dimensional Euclidean representations. We formalize the functional representation hypothesis, where data are explained through sparse compositions of structured functions. Unlike standard SAEs that represent concepts with scalar activations, SAE-NOs parameterize concepts as functions, enabling representations that capture not only a concept's presence, but also how and where it is expressed across the input domain. We achieve this through joint sparsity: concept sparsity selects active concepts, while domain sparsity governs where they are expressed. We instantiate this framework using Fourier neural operators (SAE-FNOs), parameterizing concepts as integral operators in the Fourier domain. This functional and spectral parameterization is particularly advantageous when data exhibit spatial structure across scales or when concepts are frequency-structured. We characterize SAE-FNO on vision data and demonstrate that it learns localized patterns, uses concepts more efficiently, and exhibits stable concept characteristics across sparsity levels. We further show that SAE-FNO adapts to changes in domain size and generalizes across discretizations, operating at resolutions beyond those seen during training, where standard SAEs fail. We also introduce lifting into SAEs and show theoretically and empirically that it acts as a preconditioner that accelerates optimization. Overall, our results show that moving from vector-valued to functional parameterizations, with concept and domain sparsity, extends SAEs from representing concept presence to modeling structured concept expression, highlighting the importance of parameterization.

cs.LG · cs.AI · eess.SP · stat.ML

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.09048

LeanDojo-v2: A Comprehensive Library for AI-Assisted Theorem Proving in Lean

Ryan Hsiang, William Adkisson, Robert Joseph George, Anima Anandkumar

原文摘要

Building on recent progress in machine learning for formal theorem proving in Lean, we introduce LeanDojo-v2: an end-to-end framework that unifies repository data extraction, LLM fine-tuning, proof search, AI autocompletion, and LLM-assisted theorem proving in a single software library. In addition, we provide Lean4Code, a lightweight IDE that integrates the various features of LeanDojo-v2 through a simple, mathematician-friendly UI. These tools will enable mathematicians and researchers to easily fine-tune language models on domain-specific Lean codebases and experiment with LLM-assisted theorem proving without a complex setup.

cs.LG

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.09038

Function-Space Certification for Neural Operators

Ryan Hsiang, Robert Joseph George, Anima Anandkumar

原文摘要

Certifying the robustness and physical consistency of scientific machine-learning models ensures that their predictions remain reliable beyond the test set. For neural operators, which learn PDE solution operators by mapping infinite-dimensional function spaces, certification means establishing rigorous, domain-wide guarantees for their output functions. However, standard neural network (NN) verification methods are fundamentally designed for vector-to-vector mappings, and they can only certify neural operators at fixed grid resolutions and provide no guarantees over the entire domain or across another resolution. We close this gap by introducing a function-space certification framework that verifies neural operators across the entire continuous domain. We formulate interval bound propagation (IBP) and CROWN in function space, propagating upper and lower bound functions through each layer using rigorous numerical integration. In addition, we extend bound propagation to higher-order derivatives to certify PDE residuals and solution error bounds. Experiments across several neural operator architectures and scientific problems, including the 572.5-million-parameter spherical Fourier neural operator for global weather forecasting, show that our method produces tight, domain-wide guarantees efficiently, providing a foundation for trustworthy operator learning.

cs.LG

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2512.01421

Fourier Neural Operators Explained: A Practical Perspective

Valentin Duruisseaux, Jean Kossaifi, Anima Anandkumar

原文摘要

Partial differential equations (PDEs) govern a wide variety of dynamical processes in science and engineering, yet obtaining their numerical solutions often requires high-resolution discretizations and repeated evaluations of complex operators, leading to substantial computational costs. Neural operators have recently emerged as a powerful framework for learning mappings between function spaces directly from data, enabling efficient surrogate models for PDE systems. Among these architectures, the Fourier Neural Operator (FNO) has become the most influential and widely adopted due to its elegant spectral formulation, which captures global correlations through learnable transformations in Fourier space while remaining invariant to discretization and resolution. Despite their success, the practical use of FNOs is often hindered by an incomplete understanding among practitioners of their theoretical foundations, practical constraints, and implementation details, which can lead to their incorrect or unreliable application. This work presents a comprehensive and practice-oriented guide to FNOs, unifying their mathematical principles with implementation strategies. We provide an intuitive exposition of the concepts of operator theory and signal processing that underlie the FNO, detail its spectral parameterization and the computational design of all its components, and address common misunderstandings encountered in the literature. The exposition is closely integrated with the NeuralOperator 2.0.0 library, offering modular state-of-the-art implementations that faithfully reflect the theory. By connecting rigorous foundations with practical insight, this guide aims to establish a clear and reliable framework for applying FNOs effectively across diverse scientific and engineering fields.

cs.LG

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.07646

WaSP: Wasserstein Data Synthesis Pipeline for Solver-Free Operator Learning

Ryan Liu, Chuwei Wang, Ricardo Baptista, Andrew Stuart, Anima Anandkumar

原文摘要

Neural operators offer fast surrogates for solving PDEs, but training them typically requires expensive input-solution pairs. Two solver-free alternatives include physics-informed learning and training on manufactured data. Physics-informed learning approximates solutions by minimizing PDE residuals. While the residual norm must be strong enough to control solution error, stronger norms increase mini-batch variance and hinder optimization. Manufactured data constructs input-solution pairs for learning by applying the PDE operator to artificial solutions. However, the resulting training distribution can be mutually singular with the target distribution, yielding poor accuracy on the target distribution. We introduce the Wasserstein Data Synthesis Pipeline (WaSP), which trains on manufactured input-solution pairs while gradually mitigating the distribution shift. WaSP synthesizes new pairs by applying the learned operator to a path of intermediate distributions converging to the target input distribution. Training proceeds in stages along this path. At each stage, the previous surrogate generates solutions, and the exact PDE operator derives corresponding inputs, yielding training pairs for the next surrogate. Theoretically, we prove a stepwise error-propagation bound for the distribution shift introduced during synthesis. Empirically, WaSP reduces solution error by $3.9\times$ to $9.5\times$ compared to existing solver-free methods on PDEs such as Allen-Cahn, Kuramoto-Sivashinsky, and Ginzburg-Landau, and better reproduces the two chaotic systems' statistical observables.

cs.LG · cs.AI

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.02127

Mesh2Shell Neural Operator for Inverse Shape Optimization

Xinyi Li, Xi Deng, Yubo Cao, Hudson Liu, Anima Anandkumar

原文摘要

Physics-driven design seeks high-performing geometries, which need not conform to a learned shape category. Parametric and latent-space approaches facilitate surrogate-based optimization, but restrict deformations to prescribed shape families or learned decoders. We introduce the Mesh-to-Shell Neural Operator (M2S-NO) as the mesh-to-shell component of a category-agnostic inverse shape optimization framework. We separate the explicit mesh used for design from the volumetric representation used for physics prediction. M2S-NO maps oriented surface samples to a surface-centered shell representation, such as TruncatedSDF or HeatSDF. A separate problem-specific PDE neural operator (PDE-NO) maps this representation to physical fields. Objective gradients update mesh vertices through both operators, without a latent shape decoder. Preconditioning and remeshing support free-form deformation, while fixed interfaces are controlled directly on the mesh. Optional shell-space proposals enable topology changes. We evaluate geometric matching, jet-engine brackets, and neutron moderators, with independent FEM and OpenMC evaluation. Relative to the best training designs, M2S-NO reduces maximum bracket displacement by 14.3% and increases neutron-sensor current by 13.8%. Optimization also succeeds from an out-of-distribution bracket initialization, demonstrating the framework's geometric flexibility.

cs.LG

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2610.29111

SU(2)-Equivariant Spinor Message Passing for Molecular Magnetism

Erh-Wei Sheng, Beom Seok Kang, Yuchao Lin, Danish Khan, Anima Anandkumar

原文摘要

Predicting molecular magnetic properties often requires costly electronic-structure calculations that account for spin-orbit coupling. Inspired by the spinor nature of the underlying electronic states, we introduce eSpinor, an SU(2)-equivariant graph neural network that propagates spinor features alongside the integer-rank tensors used in conventional equivariant models. We initialize spinors from eigenvectors of an approximate Hamiltonian that includes spin-orbit coupling. The network preserves their rotation laws and makes predictions independent of arbitrary eigenbasis choices. Comparisons with geometry-only and Hamiltonian-informed baselines assess the benefit of this architecture beyond supplying electronic information. On cobalt single-molecule magnet candidates, eSpinor improves magnetic anisotropy predictions, including for unseen ligands. It also improves excitation-energy predictions for dysprosium candidates with unseen ligands, extending the benefit to rotationally invariant scalar targets. For a dysprosium magnetic resonance imaging probe, it predicts real electron-spin correlations and low-field nuclear relaxation rates more accurately despite static crystal-field errors similar to the Hamiltonian-informed baseline. This dynamical advantage extends from water to methanol without retraining. These results support spinor message passing as a useful inductive bias for molecular magnetism in the tested systems and show why model selection should be guided by accuracy on the intended physical observables.

cs.LG

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arXiv ↗PDF ↗