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Neural Operators for Accelerating Scientific Simulations and Design

Kamyar Azizzadenesheli, Nikola Kovachki, Zongyi Li, Miguel Liu-Schiaffini, Jean Kossaifi, Anima Anandkumar

TL;DR

The paper tackles the bottleneck of expensive experiments and computationally intensive simulations by proposing Neural Operators that learn mappings between function spaces, enabling fast, data-driven surrogates for PDEs and spatiotemporal processes. The approach leverages discretization-convergent operator learning (e.g., Fourier Neural Operator, Graph Neural Operator, DeepONet variants) to predict solutions at arbitrary locations and resolutions, often with large speedups. Key contributions include formalization of Neural Operators, articulation of physics-informed extensions (PINO), and demonstrations across domains such as computational fluid dynamics, weather forecasting, and materials modeling, with potential for inverse design. The work argues Neural Operators can democratize scientific computation, enable rapid design and optimization, and serve as a foundation model for simulation-guided discovery, while highlighting remaining challenges in data efficiency, uncertainty quantification, and physical validity.

Abstract

Scientific discovery and engineering design are currently limited by the time and cost of physical experiments, selected mostly through trial-and-error and intuition that require deep domain expertise. Numerical simulations present an alternative to physical experiments but are usually infeasible for complex real-world domains due to the computational requirements of existing numerical methods. Artificial intelligence (AI) presents a potential paradigm shift by developing fast data-driven surrogate models. In particular, an AI framework, known as Neural Operators, presents a principled framework for learning mappings between functions defined on continuous domains, e.g., spatiotemporal processes and partial differential equations (PDE). They can extrapolate and predict solutions at new locations unseen during training, i.e., perform zero-shot super-resolution. Neural Operators can augment or even replace existing simulators in many applications, such as computational fluid dynamics, weather forecasting, and material modeling, while being 4-5 orders of magnitude faster. Further, Neural Operators can be integrated with physics and other domain constraints enforced at finer resolutions to obtain high-fidelity solutions and good generalization. Since Neural Operators are differentiable, they can directly optimize parameters for inverse design and other inverse problems. We believe that Neural Operators present a transformative approach to simulation and design, enabling rapid research and development.

Neural Operators for Accelerating Scientific Simulations and Design

TL;DR

The paper tackles the bottleneck of expensive experiments and computationally intensive simulations by proposing Neural Operators that learn mappings between function spaces, enabling fast, data-driven surrogates for PDEs and spatiotemporal processes. The approach leverages discretization-convergent operator learning (e.g., Fourier Neural Operator, Graph Neural Operator, DeepONet variants) to predict solutions at arbitrary locations and resolutions, often with large speedups. Key contributions include formalization of Neural Operators, articulation of physics-informed extensions (PINO), and demonstrations across domains such as computational fluid dynamics, weather forecasting, and materials modeling, with potential for inverse design. The work argues Neural Operators can democratize scientific computation, enable rapid design and optimization, and serve as a foundation model for simulation-guided discovery, while highlighting remaining challenges in data efficiency, uncertainty quantification, and physical validity.

Abstract

Scientific discovery and engineering design are currently limited by the time and cost of physical experiments, selected mostly through trial-and-error and intuition that require deep domain expertise. Numerical simulations present an alternative to physical experiments but are usually infeasible for complex real-world domains due to the computational requirements of existing numerical methods. Artificial intelligence (AI) presents a potential paradigm shift by developing fast data-driven surrogate models. In particular, an AI framework, known as Neural Operators, presents a principled framework for learning mappings between functions defined on continuous domains, e.g., spatiotemporal processes and partial differential equations (PDE). They can extrapolate and predict solutions at new locations unseen during training, i.e., perform zero-shot super-resolution. Neural Operators can augment or even replace existing simulators in many applications, such as computational fluid dynamics, weather forecasting, and material modeling, while being 4-5 orders of magnitude faster. Further, Neural Operators can be integrated with physics and other domain constraints enforced at finer resolutions to obtain high-fidelity solutions and good generalization. Since Neural Operators are differentiable, they can directly optimize parameters for inverse design and other inverse problems. We believe that Neural Operators present a transformative approach to simulation and design, enabling rapid research and development.
Paper Structure (5 sections, 2 equations, 3 figures)

This paper contains 5 sections, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Neural Operators
  3. Motivation
  4. Formulation
  5. Challenges and Outlook

Figures (3)

  • Figure 1: Left: The x-axis is the Fourier wavenumber and y-axis is the energy per spectrum. Fourier Neural Operators (FNO) can extrapolate to unseen frequencies in Kolmogorov Flows pino using only limited resolution training data. Physics-informed Neural Operator (PINO) uses both training data and the PDE equation for the loss function, and can perfectly recover the ground-truth spectrum. A trained UNet with trilinear interpolation (NN+Interpolation) has severe distortions at higher frequencies, beyond the resolution of training data. Right: The x-axis is the resolution of the test data, and y-axis is the test error at that given resolution. Neural Operators are discretization convergent, meaning the model converges to the target continuum operator as the discretization is refined. On the Darcy equation, we train each fixed architecture UNet, FNO, and Graph Neural Operator (GNO) at a given resolution and test at that same resolution kovachki2023neural (no super-resolution). As shown in the figure, FNO and GNO have consistent errors as resolution increases, but UNet has increasing errors since the size of its receptive field changes with resolution, and it does not enjoy the guarantees of discretization convergence. UNet is only able to maintain the same test error as resolution increases if the number of parameters increases. In this case, we increase the convolutional filter size, corresponding to about 2.2M, 6.0M, 11.8M, and 19.3M parameters, respectively.
  • Figure 2: Comparison of Neural Networks (NNs) with Neural Operators (NOs).
  • Figure 3: Diagram comparing pseudo-spectral solver, Fourier Neural Operator (FNO), and the general Neural Operator architecture. FT and IFT refer to Fourier and Inverse Fourier Transforms. In general, lifting and projection operators $\mathcal{P}$, $\mathcal{Q}$ can be non-linear. Pseudo-spectral solvers are popular numerical solvers for fluid dynamics where the Fourier basis is utilized, and operations are iteratively carried out, as shown. The Fourier Neural Operator (FNO) is inspired by the pseudo-spectral solver, but has a non-linear representation that is learned. FNO is a special case of the Neural-Operator framework, shown in the last row, where the kernel integration can be carried out through different methods, e.g., direct discretization or through Fourier transform.