Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs

Enabling Automatic Differentiation with Mollified Graph Neural Operators

Physics-informed neural operators offer a powerful framework for learning solution operators of partial differential equations (PDEs) by combining data and physics losses. However, these physics losses rely on derivatives. Computing these derivatives remains challenging, with spectral and finite difference methods introducing approximation errors due to finite resolution. Here, we propose the mollified graph neural operator (mGNO), the first method to leverage automatic differentiation and compute \emph{exact} gradients on arbitrary geometries. This enhancement enables efficient training on irregular grids and varying geometries while allowing seamless evaluation of physics losses at randomly sampled points for improved generalization. For a PDE example on regular grids, mGNO paired with autograd reduced the L2 relative data error by 20x compared to finite differences, although training was slower. It can also solve PDEs on unstructured point clouds seamlessly, using physics losses only, at resolutions vastly lower than those needed for finite differences to be accurate enough. On these unstructured point clouds, mGNO leads to errors that are consistently 2 orders of magnitude lower than machine learning baselines (Meta-PDE) for comparable runtimes, and also delivers speedups from 1 to 3 orders of magnitude compared to the numerical solver for similar accuracy. mGNOs can also be used to solve inverse design and shape optimization problems on complex geometries.

Factorized Implicit Global Convolution for Automotive Computational Fluid Dynamics Prediction

Computational Fluid Dynamics (CFD) is crucial for automotive design, requiring the analysis of large 3D point clouds to study how vehicle geometry affects pressure fields and drag forces. However, existing deep learning approaches for CFD struggle with the computational complexity of processing high-resolution 3D data. We propose Factorized Implicit Global Convolution (FIGConv), a novel architecture that efficiently solves CFD problems for very large 3D meshes with arbitrary input and output geometries. FIGConv achieves quadratic complexity $O(N^2)$, a significant improvement over existing 3D neural CFD models that require cubic complexity $O(N^3)$. Our approach combines Factorized Implicit Grids to approximate high-resolution domains, efficient global convolutions through 2D reparameterization, and a U-shaped architecture for effective information gathering and integration. We validate our approach on the industry-standard Ahmed body dataset and the large-scale DrivAerNet dataset. In DrivAerNet, our model achieves an $R^2$ value of 0.95 for drag prediction, outperforming the previous state-of-the-art by a significant margin. This represents a 40% improvement in relative mean squared error and a 70% improvement in absolute mean squared error over previous methods.

Tensor-GaLore: Memory-Efficient Training via Gradient Tensor Decomposition

We present Tensor-GaLore, a novel method for efficient training of neural networks with higher-order tensor weights. Many models, particularly those used in scientific computing, employ tensor-parameterized layers to capture complex, multidimensional relationships. When scaling these methods to high-resolution problems makes memory usage grow intractably, and matrix based optimization methods lead to suboptimal performance and compression. We propose to work directly in the high-order space of the complex tensor parameter space using a tensor factorization of the gradients during optimization. We showcase its effectiveness on Fourier Neural Operators (FNOs), a class of models crucial for solving partial differential equations (PDE) and prove the theory of it. Across various PDE tasks like the Navier Stokes and Darcy Flow equations, Tensor-GaLore achieves substantial memory savings, reducing optimizer memory usage by up to 75%. These substantial memory savings across AI for science demonstrate Tensor-GaLore's potential.

A Library for Learning Neural Operators

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.

Exploring the design space of deep-learning-based weather forecasting systems

Despite tremendous progress in developing deep-learning-based weather forecasting systems, their design space, including the impact of different design choices, is yet to be well understood. This paper aims to fill this knowledge gap by systematically analyzing these choices including architecture, problem formulation, pretraining scheme, use of image-based pretrained models, loss functions, noise injection, multi-step inputs, additional static masks, multi-step finetuning (including larger stride models), as well as training on a larger dataset. We study fixed-grid architectures such as UNet, fully convolutional architectures, and transformer-based models, along with grid-invariant architectures, including graph-based and operator-based models. Our results show that fixed-grid architectures outperform grid-invariant architectures, indicating a need for further architectural developments in grid-invariant models such as neural operators. We therefore propose a hybrid system that combines the strong performance of fixed-grid models with the flexibility of grid-invariant architectures. We further show that multi-step fine-tuning is essential for most deep-learning models to work well in practice, which has been a common practice in the past. Pretraining objectives degrade performance in comparison to supervised training, while image-based pretrained models provide useful inductive biases in some cases in comparison to training the model from scratch. Interestingly, we see a strong positive effect of using a larger dataset when training a smaller model as compared to training on a smaller dataset for longer. Larger models, on the other hand, primarily benefit from just an increase in the computational budget. We believe that these results will aid in the design of better weather forecasting systems in the future.

Fourier Neural Operators for Learning Dynamics in Quantum Spin Systems

Fourier Neural Operators (FNOs) excel on tasks using functional data, such as those originating from partial differential equations. Such characteristics render them an effective approach for simulating the time evolution of quantum wavefunctions, which is a computationally challenging, yet coveted task for understanding quantum systems. In this manuscript, we use FNOs to model the evolution of random quantum spin systems, so chosen due to their representative quantum dynamics and minimal symmetry. We explore two distinct FNO architectures and examine their performance for learning and predicting time evolution using both random and low-energy input states. Additionally, we apply FNOs to a compact set of Hamiltonian observables ($\sim\text{poly}(n)$) instead of the entire $2^n$ quantum wavefunction, which greatly reduces the size of our inputs and outputs and, consequently, the requisite dimensions of the resulting FNOs. Moreover, this Hamiltonian observable-based method demonstrates that FNOs can effectively distill information from high-dimensional spaces into lower-dimensional spaces. The extrapolation of Hamiltonian observables to times later than those used in training is of particular interest, as this stands to fundamentally increase the simulatability of quantum systems past both the coherence times of contemporary quantum architectures and the circuit-depths of tractable tensor networks.

Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs

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 lack of large 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 the function space. 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 and fluid-structure interactions, we found CoDA-NO to outperform existing methods on the few-shot learning task by over $36\%$. The code is available at https://github.com/ashiq24/CoDA-NO.

Equivariant Graph Neural Operator for Modeling 3D Dynamics

Modeling the complex three-dimensional (3D) dynamics of relational systems is an important problem in the natural sciences, with applications ranging from molecular simulations to particle mechanics. Machine learning methods have achieved good success by learning graph neural networks to model spatial interactions. However, these approaches do not faithfully capture temporal correlations since they only model next-step predictions. In this work, we propose Equivariant Graph Neural Operator (EGNO), a novel and principled method that directly models dynamics as trajectories instead of just next-step prediction. Different from existing methods, EGNO explicitly learns the temporal evolution of 3D dynamics where we formulate the dynamics as a function over time and learn neural operators to approximate it. To capture the temporal correlations while keeping the intrinsic SE(3)-equivariance, we develop equivariant temporal convolutions parameterized in the Fourier space and build EGNO by stacking the Fourier layers over equivariant networks. EGNO is the first operator learning framework that is capable of modeling solution dynamics functions over time while retaining 3D equivariance. Comprehensive experiments in multiple domains, including particle simulations, human motion capture, and molecular dynamics, demonstrate the significantly superior performance of EGNO against existing methods, thanks to the equivariant temporal modeling.

Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs

Memory complexity and data scarcity have so far prohibited learning solution operators of partial differential equations (PDEs) at high resolutions. We address these limitations by introducing a new data efficient and highly parallelizable operator learning approach with reduced memory requirement and better generalization, called multi-grid tensorized neural operator (MG-TFNO). MG-TFNO scales to large resolutions by leveraging local and global structures of full-scale, real-world phenomena, through a decomposition of both the input domain and the operator's parameter space. Our contributions are threefold: i) we enable parallelization over input samples with a novel multi-grid-based domain decomposition, ii) we represent the parameters of the model in a high-order latent subspace of the Fourier domain, through a global tensor factorization, resulting in an extreme reduction in the number of parameters and improved generalization, and iii) we propose architectural improvements to the backbone FNO. Our approach can be used in any operator learning setting. We demonstrate superior performance on the turbulent Navier-Stokes equations where we achieve less than half the error with over 150x compression. The tensorization combined with the domain decomposition, yields over 150x reduction in the number of parameters and 7x reduction in the domain size without losses in accuracy, while slightly enabling parallelism.