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Anima Anandkumar

Publications and source records attributed to Anima Anandkumar.

At least 19 recordsLinked to original sources
arXiv2026

Mesh2Shell Neural Operator for Inverse Shape Optimization

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↗
arXiv2026

A Multi-Instance LLM Framework for Rating the Quality of Surgical Feedback

Verbal feedback delivered by attending surgeons in the operating room plays a critical formative role in resident trainee skill acquisition. Yet, assessing the quality of trainer feedback and its effectiveness in influencing trainee behavior during live surgery remains a challenge. Prior studies assessed feedback content relying on extensive manual annotation by expert human raters and focused on developing broad taxonomies that overlook the qualitative aspects of feedback delivery such as clarity or urgency. Limited existing automated methods, including keyword analysis and topic modeling, also fail to capture these nuanced aspects. We introduce a two-stage LLM-based framework that discovers interpretable feedback quality criteria grounded in the context of surgical training. Our method uses multi-instance prompting via repeated independent sampling of candidate criteria followed by consolidation, and surgical domain knowledge injection to discover a small set of human interpretable scoring criteria (e.g., Encouragement, Urgency, Clarity). These criteria are then used to automatically score live surgical feedback via an LLM-as-a-judge approach. Evaluation on 4.2k trainer feedback instances demonstrates that our AI-discovered criteria outperform prior content-based frameworks in predicting feedback effectiveness, including observed trainee behavioral adjustments and trainer approval. This work advances scalable, human-aligned assessment of communication quality in the operating room and provides a foundation for improving surgical teaching practices.

cs.CL↗
arXiv2026

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

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.

physics.chem-ph↗
arXiv2026

Performance of Neural Operator Baselines on 'The Well' Dataset

The Well, a 15-terabyte dataset spanning 16 physical simulation domains, provides an extensive benchmark for developing and evaluating machine learning surrogate models for complex physical systems. The baselines released with the benchmark were presented by their authors as illustrative examples obtained at small computational cost, trained with zero to very limited hyperparameter tuning and under a highly constraining 12-hour per-run wall-clock limit. These results have subsequently been treated as fully tuned benchmarks, and in some cases effectively interpreted as unconstrained performance baselines. We perform a short non-exhaustive hyperparameter search for the Tucker-factorized Fourier Neural Operator (TFNO) under the same 12-hour per-run wall-clock limit, while using only a single input timestep rather than the history of up to four used by the published baselines. Despite this more restrictive setup, tuning alone reduces VRMSE below both published FNO and TFNO baselines on 15 of 17 tasks, sometimes by several orders of magnitude. This sensitivity to a modest tuning effort underscores that the released neural operator baselines should not be treated as architectural benchmarks.

cs.LG↗
arXiv2026

FG-ConvNO: A Geometry-Aware Neural Operator for Propeller CFD Prediction

Computational Fluid Dynamics (CFD) is essential in contemporary engineering design, yet it remains expensive to compute and can take thousands of CPU/GPU hours or more for a single high-fidelity simulation. Recent developments in machine learning for scientific modeling have led to powerful surrogate frameworks that can approximate flow fields and physical quantities by using a fraction of the computational cost. However, most of these methods require largescale homogeneous datasets or simple geometries, exhibiting severe generalization gaps when applied to complex and data-scarce configurations. To address this, we propose the Factorized Geometry Convolutional Neural Operator (FG-ConvNO) for robust geometry-aware propeller pressure prediction under limited training data. We introduce geometryaware encoding, efficient interpolation-based decoding, and auxiliary global quantity supervision. In addition, we incorporate DISCO blocks that use discrete continuous convolutions to make the model discretization agnostic. In our experiments, FG-ConvNO achieves the highest accuracy for both surface pressure and global thrust prediction on a procedurally generated propeller CFD dataset. Geometry-aware encoding and architectural simplification improve stability, while random vertex subsampling enhances robustness. Overall, adding geometric priors to grid-based convolutional operators reduces overfitting on complex propeller geometries and improves generalization in low-data regimes.

cs.LG↗
arXiv2026

FC-PINO: High Precision Physics-Informed Neural Operators via Fourier Continuation

The physics-informed neural operator (PINO) is a machine learning paradigm that has demonstrated promising results for learning solutions to partial differential equations (PDEs). It leverages the Fourier Neural Operator to learn solution operators in function spaces and leverages physics losses during training to penalize deviations from known physics laws. Spectral differentiation provides an efficient way to compute derivatives for the physics losses, but it inherently assumes periodicity. When applied to non-periodic functions, this assumption can lead to significant errors, including Gibbs phenomena near domain boundaries which degrade the accuracy of both function representations and derivative computations. To overcome this limitation, we introduce the FC-PINO (Fourier-Continuation-based Physics-Informed Neural Operator) architecture which extends the accuracy and efficiency of PINO and spectral differentiation to non-periodic and non-smooth PDEs. In FC-PINO, we propose integrating Fourier continuation into the PINO framework, and test two different continuation approaches: FC-Legendre and FC-Gram. By transforming non-periodic signals into periodic functions on extended domains in a well-conditioned manner, Fourier continuation enables fast and accurate derivative computations. This approach avoids the discretization sensitivity of finite differences and the memory overhead of automatic differentiation. We demonstrate that standard PINO fails (without padding) or struggles (even with padding) to solve non-periodic and non-smooth PDEs with high precision, across challenging benchmarks. In contrast, the proposed FC-PINO provides accurate, robust, and scalable solutions, substantially outperforming PINO alternatives, and demonstrating that Fourier continuation is critical for extending PINO to a wider range of PDE problems when high-precision solutions are needed.

cs.LG↗
arXiv2026

MLIPDiff: Guiding Denoising Diffusion Models for Structure-Based Drug Design via Machine-Learned Interatomic Potentials

Diffusion models have become a dominant approach to structure-based drug design. However, because they are trained primarily to fit reference structures, they often place ligands in physically implausible poses that clash with the protein. We introduce MLIPDiff, an inference-time guidance method that can be applied to pretrained diffusion models for ligand generation. The method uses physics-informed guidance from UMA (Universal Models for Atoms), a machine-learned interatomic potential (MLIP). The guidance combines four weighted energy terms (protein-ligand interaction, ligand strain, steric clash, and pocket anchoring) and uses the original model checkpoint without retraining. On CrossDocked2020, MLIPDiff completely eliminates protein-ligand collisions while preserving the ligand's local geometry, improving bond-length distributions and physical plausibility relative to the base model. Compared to models such as NucleusDiff that are specifically designed to reduce atomic clashes, MLIPDiff achieves clash-free structures without compromising local geometry. It maintains the base model's PoseBusters pass rate and attains the best median Vina Score, Vina Min, and Vina Dock, surpassing both the base model and NucleusDiff, with a median of 59.6\% of ligands per pocket binding more strongly than the reference. MLIPDiff thus addresses a limitation of prior methods by generating physically plausible structures with favorable predicted binding affinity using a single frozen model and no additional training.

cs.LG↗
arXiv2026

Planning under Noise with Neural Operators

Planning under uncertain control execution noise requires reliable feedback policies across diverse environments. We introduce Planning Under Noise (PUN), a neural operator that learns noise-aware value fields for planning in unseen geometries. Building on an established screened-Poisson formulation of stochastic optimal control, we address two challenges: specifying boundary conditions that support navigation across varying geometries and efficiently computing fields with reliable gradients. Zero-Dirichlet obstacle boundaries terminate sampled paths at walls, weakening the goal signal along long routes and making it harder to estimate reliably. We instead impose reflecting Neumann conditions on obstacles and a Dirichlet condition at the goal, allowing sampled paths to continue exploring free space until they reach the target. To solve the resulting PDE without a configuration-space mesh, we extend a Monte Carlo (MC) solver, Walk on Stars (WoSt), to higher dimensions. However, local MC fluctuations can reverse estimated gradient directions and cause planning failures, making direct planning require costly sampling. PUN amortizes this computation by learning geometry-to-field mappings from noisy, partially converged Monte Carlo labels, avoiding expensive convergence for every training example. At inference, test-time scaling uses bounded local search guided by the predicted global field. We evaluate PUN on planning tasks across dimensions. Across four noise levels, PUN matches or outperforms all compared planners, achieving 95-100% success on the 4D replanning benchmark while maintaining the lowest runtime. At the highest tested noise level in 4D, PUN achieves 10.7-60.6% relative gains in success rate over other methods while running 3.29-32.15 times faster.

cs.LG↗
arXiv2026

Physics is the Best Teacher: Consistency Learning for Time-Invariant Operators of Chaotic Dynamics

Accelerating the prediction of long-term behavior in chaotic systems is crucial in scientific computing. However, existing methods rely on numerical solvers or autoregressive models that advance one small step at a time, which makes long horizons expensive. We instead view this problem as learning the system's time-invariant evolution operator, which jumps the state across a large time span in a single evaluation. To this end, we derive the consistency equations a time-invariant operator must satisfy, with differential and compositional objectives in physical time. These equations also connect the learned operator to the physics-prescribed instant dynamics, enabling physics embedding in consistency learning. Across five chaotic systems, we find that physics-distilled consistency makes both short-term trajectories and long-term statistics more accurate. The learned operator survives temporal extrapolation and requires one-tenth as many evaluations as autoregressive rollout, offering an efficient route to long-term simulation of chaotic dynamics.

cs.LG↗
arXiv2026

Neural Operator Splitting for Fluid Dynamics in Complex Geometry

Solving dynamical systems in complex and diverse geometries remains challenging for both numerical solvers and learning-based surrogate models. We propose Neural Operator Splitting, a framework for learning fluid dynamics while preserving the physical structure of fractional-step numerical schemes. Instead of learning the full-step evolution monolithically, we decompose each time step into physically interpretable steps, and learn each operator separately using intermediate states from the underlying numerical solver. Guided by the linearity properties of the fractional steps, we introduce a conditional-linear Fourier neural operator (CL-FNO) whose linear state mappings are modulated by nonlinear conditioning on inputs such as boundary geometry. We further introduce two complementary strategies, symmetry sampling and corrective cycling, to mitigate error accumulation during autoregressive rollout. Experiments on 2D and 3D fluid flows with varying geometries and initial conditions show that our full model reduces mean relative $L^2$ rollout error by 68\% in 2D and 40\% in 3D compared with a single full-step neural operator. At comparable error levels, neural-operator inference is approximately $10^3$ times faster per step than the Monte Carlo solver.

cs.LG↗
arXiv2026

Learnable Splines for Unstructured Neural Operators

Neural operators approximate solution operators of partial differential equations at a fraction of the cost of numerical solvers. Efficient grid-based architectures such as the Fourier neural operator (FNO) assume uniform rectangular grids, whereas engineering simulations use large, irregular, and locally refined meshes. We introduce L-SplineFNO, a memory-efficient neural operator that couples an FNO backbone with a sparse, learnable tensor-product B-spline transfer between unstructured meshes and a latent Cartesian grid. A local normalised scatter avoids the geometry-dependent solve of analytic spline transformation while preserving compact support, partition of unity, and spline smoothness by construction. The neural operator, conditioned on global design and operating parameters, learns the mapping of geometry to the field solution. Across six three-dimensional engineering benchmarks, L-SplineFNO achieves strong predictive accuracy while requiring substantially less memory and compute than baseline operators, scaling to 26.6 million nodes in training and 46 million in inference on a single A100 GPU. It also supports zero-shot super-resolution and efficient closed-form spatial derivatives. Uniquely, the analytic spline decoder yields rigorous prediction bounds over continuous space and uncertain parameters, via Bernstein convex hulls and Taylor models, demonstrated on a 29.2-million-node MAST-U tokamak model.

cs.AI↗
arXiv2026

Beyond DFT electronic band structures via dual-space orbital learning

Accurate electronic band structures are essential for predicting material properties, but high-fidelity calculations are computationally demanding. We introduce O-kRystal, a dual-space orbital-learning framework that uses a lower-cost Perdew-Burke-Ernzerhof (PBE) calculation as both an energy baseline and a source of structured orbital information. Combining PBE band energies with real- and reciprocal-space orbital matrices, O-kRystal predicts higher-level corrections for individual electronic states as a function of crystal momentum and spin. For states within 5 eV of the band-edge midpoint, it predicts Heyd-Scuseria-Ernzerhof (HSE06) band energies with a mean absolute error of $0.171 \pm 0.009$ eV across nearly 6,000 three-dimensional materials spanning insulating, metallic, magnetic, and nonmagnetic systems. On a separate benchmark of six strongly correlated materials, it reduces the corresponding PBE band-energy MAE by 68.7% and resolves orbital-dependent changes in the relative placement of transition-metal d and ligand p states. Despite training only at high-symmetry points, it maintains comparable accuracy at unseen points along band paths and reconstructs HSE06 densities of states from dense Brillouin-zone predictions. Finally, training O-kRystal on $G_0W_0$ quasiparticle energies for two-dimensional materials demonstrates its applicability across levels of theory and the potential of orbital learning for higher-fidelity, momentum-resolved electronic-structure prediction beyond density functional theory.

cs.LG↗
arXiv2026

PTNO: Training Neural Operators with Noisy Monte Carlo Estimates for Particle Transport Problems

Particle transport under multiple scattering is central to radiative transfer and plasma physics, yet high-fidelity Monte Carlo (MC) simulations must trace prohibitively many particles. Learning-based surrogates can amortize this cost, but typically train on expensive, well-converged MC solutions. We propose the Particle Transport Neural Operator (PTNO), a neural operator that learns particle transport surrogates directly from noisy, low-cost MC labels. Such labels pose two challenges: (1) high variance, which destabilizes standard supervised learning, and (2) a high dynamic range (HDR) spanning many orders of magnitude. For the first, we learn the solution operator from noisy labels of many configurations, amortizing MC cost and generalizing to unseen configurations. Because MC labels are unbiased, we show that the squared loss on them shares its minimizer with the loss on converged solutions, and our budget-allocation study over training scenes $M$, MC samples per render $N$, and independent renders per scene $K$ shows that many noisy scenes beat fewer converged ones. For the second, a nonlinear transform such as the logarithm biases noisy supervision. Instead, PTNO keeps labels in physical space and enforces positivity with a softplus output layer that represents small values effectively. We further train with a pointwise relative $L_2$ loss (PRelL2), the stop-gradient relative loss of HDR denoising and neural rendering, which normalizes each residual by the stop-gradient prediction instead of the noisy label. We demonstrate PTNO on neutron transport in fusion reactors and radiative transfer in participating media. On the two neutronics tasks, PTNO is $10^4$-$10^5\times$ faster than converged MC on the same CPU and $10^3$-$10^5\times$ cheaper than MC at matched accuracy; on the two radiative-transfer tasks, MC at matched accuracy costs $0.8$-$11\times$ as much as PTNO.

cs.AI↗
arXiv2026

A library for differentiable signal processing and machine learning on the sphere

The two-dimensional sphere embedded in three-dimensional Euclidean space S2, plays a central role in a variety of scientific and engineering domains, including geophysics, planetary science, geodesy, atmospheric physics, quantum chemistry, cosmology, and virtual reality, among many others. As machine learning increasingly permeates these fields, the demand grows for robust tools that process and model functions on the sphere, while respecting the inherent topological and symmetry properties of the domain. We present torch-harmonics, a comprehensive library that offers efficient, differentiable implementations of advanced signal processing and machine learning (ML) methods for spherical data. These include the spherical harmonic transform (SHT), the spherical analogue of the Fourier transform, vector spherical harmonics, discrete-continuous and spectral convolutions, as well as both global and neighborhood spherical attention mechanisms. Beyond traditional representations, torch-harmonics provides the building blocks for state-of-the-art spherical ML architectures such as spherical transformers in order to enable scalable, rotationally-aware learning and inference in modern scientific and engineering applications.

cs.LG↗
arXiv2026

Counterfactual Predictions in Scientific Emulators Without Controlled Experiments

Many scientific questions require reasoning about what was never observed: What if the conditions, interventions, or history had been different? Models can predict accurately on observed data yet fail on such what-if queries when correlated inputs are varied independently. A common remedy is to add controlled simulation data in which these factors are explicitly disentangled, but this requires access to a simulator, can be computationally expensive, and inherits the simulator's modeling assumptions. We introduce ReRoute, a framework for targeted scientific what-if prediction that combines factual data with partial mechanistic knowledge, without requiring controlled intervention data for adaptation. ReRoute fixes the queried input of a pretrained backbone to a reference value, reintroduces its variation through a known mechanistic pathway, and fine-tunes on the original factual data, while leaving downstream effects to the learned dynamics. We provide a causal identification result for this construction under explicit structural assumptions, with the core argument machine-checked in Lean. After showing that ReRoute achieves highly accurate counterfactual predictions in a controlled advection-diffusion system where exact responses are available, we turn to state-of-the-art climate emulation. On held-out coupled-climate interventions, ReRoute reduces aggregate climate error by 18.2-31.8% under severe CO$_2$ distribution shifts while preserving skill under standard conditions, at a small fraction of the cost of retraining on additional controlled simulations, without even accounting for the substantial expense of generating such data. Finally, on an emulator trained from historical ERA5 reanalysis, where no counterfactual reference exists, ReRoute preserves substantially more of the surface warming implied by the observed boundary conditions under a fixed-CO$_2$ counterfactual.

cs.LG↗
arXiv2026

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

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 information is rarely available when a mission must be planned. Computational fluid dynamics (CFD) can produce 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 queryable 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 seconds, compared with roughly 8 hours 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.

cs.AI↗
arXiv2026

Stability Framework for the Singularity of the Euler Equations on $\mathbb{R}^3$

In a recent numerical study, we found a high-precision singular profile for the Euler equations on the unbounded domain $\mathbb{R}^3$. The present manuscript complements that study by establishing in detail a preliminary framework for proving (nonlinear) stability of the approximate self-similar profile, reducing the analysis to a large but finite collection of explicit estimates and computable constants. Conditional on rigorous certification of the estimates and constants appearing in the argument, and on the candidate profile satisfying the required nonlinear stability conditions, the framework closes the stability proof and, crucially, allows the resulting stable rescaled profile to be reconstructed as an admissible solution in the original variables that becomes singular in finite physical time. With the overall stability and reconstruction mechanisms formulated, the remaining work within this approach is largely quantitative: determining whether the explicit constants and margins can be rigorously certified with sufficient positive margin and, where necessary, sharpening selected analytic estimates.

math.AP↗
arXiv2026

Self-Similar Singularity of the Euler Equations on $\mathbb{R}^3$

We provide evidence of a finite-time singularity in the 3D Euler equations on the unbounded domain. Using a physics-informed neural network (PINN) with a self-similar ansatz, we find an approximate singular profile for the Euler system at the critical blowup rate of $0.5$ and certify it using a spline representation. The transport field associated with the obtained profile has local outgoing property throughout the domain that suggests linear damping, a key stabilizing mechanism for the candidate profile. We also establish a framework for proving nonlinear stability of the approximate self-similar profile, reducing the analysis to a large but finite collection of explicit estimates and computable constants.

math.AP↗