Maximal Update Parametrization and Zero-Shot Hyperparameter Transfer for Fourier Neural Operators

Fourier Neural Operators (FNOs) offer a principled approach for solving complex partial differential equations (PDEs). However, scaling them to handle more complex PDEs requires increasing the number of Fourier modes, which significantly expands the number of model parameters and makes hyperparameter tuning computationally impractical. To address this, we introduce $\mu$Transfer-FNO, a zero-shot hyperparameter transfer technique that enables optimal configurations, tuned on smaller FNOs, to be directly applied to billion-parameter FNOs without additional tuning. Building on the Maximal Update Parametrization ($\mu$P) framework, we mathematically derive a parametrization scheme that facilitates the transfer of optimal hyperparameters across models with different numbers of Fourier modes in FNOs, which is validated through extensive experiments on various PDEs. Our empirical study shows that Transfer-FNO reduces computational cost for tuning hyperparameters on large FNOs while maintaining or improving accuracy.

From Theory to Application: A Practical Introduction to Neural Operators in Scientific Computing

This focused review explores a range of neural operator architectures for approximating solutions to parametric partial differential equations (PDEs), emphasizing high-level concepts and practical implementation strategies. The study covers foundational models such as Deep Operator Networks (DeepONet), Principal Component Analysis-based Neural Networks (PCANet), and Fourier Neural Operators (FNO), providing comparative insights into their core methodologies and performance. These architectures are demonstrated on two classical linear parametric PDEs: the Poisson equation and linear elastic deformation. Beyond forward problem-solving, the review delves into applying neural operators as surrogates in Bayesian inference problems, showcasing their effectiveness in accelerating posterior inference while maintaining accuracy. The paper concludes by discussing current challenges, particularly in controlling prediction accuracy and generalization. It outlines emerging strategies to address these issues, such as residual-based error correction and multi-level training. This review can be seen as a comprehensive guide to implementing neural operators and integrating them into scientific computing workflows.

Graph Fourier Neural Kernels (G-FuNK): Learning Solutions of Nonlinear Diffusive Parametric PDEs on Multiple Domains

Predicting time-dependent dynamics of complex systems governed by non-linear partial differential equations (PDEs) with varying parameters and domains is a challenging task motivated by applications across various fields. We introduce a novel family of neural operators based on our Graph Fourier Neural Kernels, designed to learn solution generators for nonlinear PDEs in which the highest-order term is diffusive, across multiple domains and parameters. G-FuNK combines components that are parameter- and domain-adapted with others that are not. The domain-adapted components are constructed using a weighted graph on the discretized domain, where the graph Laplacian approximates the highest-order diffusive term, ensuring boundary condition compliance and capturing the parameter and domain-specific behavior. Meanwhile, the learned components transfer across domains and parameters via Fourier Neural Operators. This approach naturally embeds geometric and directional information, improving generalization to new test domains without need for retraining the network. To handle temporal dynamics, our method incorporates an integrated ODE solver to predict the evolution of the system. Experiments show G-FuNK's capability to accurately approximate heat, reaction diffusion, and cardiac electrophysiology equations across various geometries and anisotropic diffusivity fields. G-FuNK achieves low relative errors on unseen domains and fiber fields, significantly accelerating predictions compared to traditional finite-element solvers.

Spectral Convolutional Conditional Neural Processes

Conditional Neural Processes (CNPs) constitute a family of probabilistic models that harness the flexibility of neural networks to parameterize stochastic processes. Their capability to furnish well-calibrated predictions, combined with simple maximum-likelihood training, has established them as appealing solutions for addressing various learning problems, with a particular emphasis on meta-learning. A prominent member of this family, Convolutional Conditional Neural Processes (ConvCNPs), utilizes convolution to explicitly introduce translation equivariance as an inductive bias. However, ConvCNP's reliance on local discrete kernels in its convolution layers can pose challenges in capturing long-range dependencies and complex patterns within the data, especially when dealing with limited and irregularly sampled observations from a new task. Building on the successes of Fourier neural operators (FNOs) for approximating the solution operators of parametric partial differential equations (PDEs), we propose Spectral Convolutional Conditional Neural Processes (SConvCNPs), a new addition to the NPs family that allows for more efficient representation of functions in the frequency domain.

Neural Green's Operators for Parametric Partial Differential Equations

This work introduces neural Green's operators (NGOs), a novel neural operator network architecture that learns the solution operator for a parametric family of linear partial differential equations (PDEs). Our construction of NGOs is derived directly from the Green's formulation of such a solution operator. Similar to deep operator networks (DeepONets) and variationally mimetic operator networks (VarMiONs), NGOs constitutes an expansion of the solution to the PDE in terms of basis functions, that is returned from a sub-network, contracted with coefficients, that are returned from another sub-network. However, in accordance with the Green's formulation, NGOs accept weighted averages of the input functions, rather than sampled values thereof, as is the case in DeepONets and VarMiONs. Application of NGOs to canonical linear parametric PDEs shows that, while they remain competitive with DeepONets, VarMiONs and Fourier neural operators when testing on data that lie within the training distribution, they robustly generalize when testing on finer-scale data generated outside of the training distribution. Furthermore, we show that the explicit representation of the Green's function that is returned by NGOs enables the construction of effective preconditioners for numerical solvers for PDEs.

Learning Flame Evolution Operator under Hybrid Darrieus Landau and Diffusive Thermal Instability

Recent advancements in the integration of artificial intelligence (AI) and machine learning (ML) with physical sciences have led to significant progress in addressing complex phenomena governed by nonlinear partial differential equations (PDE). This paper explores the application of novel operator learning methodologies to unravel the intricate dynamics of flame instability, particularly focusing on hybrid instabilities arising from the coexistence of Darrieus-Landau (DL) and Diffusive-Thermal (DT) mechanisms. Training datasets encompass a wide range of parameter configurations, enabling the learning of parametric solution advancement operators using techniques such as parametric Fourier Neural Operator (pFNO), and parametric convolutional neural networks (pCNN). Results demonstrate the efficacy of these methods in accurately predicting short-term and long-term flame evolution across diverse parameter regimes, capturing the characteristic behaviors of pure and blended instabilities. Comparative analyses reveal pFNO as the most accurate model for learning short-term solutions, while all models exhibit robust performance in capturing the nuanced dynamics of flame evolution. This research contributes to the development of robust modeling frameworks for understanding and controlling complex physical processes governed by nonlinear PDE.

Parametric Learning of Time-Advancement Operators for Unstable Flame Evolution

This study investigates the application of machine learning, specifically Fourier Neural Operator (FNO) and Convolutional Neural Network (CNN), to learn time-advancement operators for parametric partial differential equations (PDEs). Our focus is on extending existing operator learning methods to handle additional inputs representing PDE parameters. The goal is to create a unified learning approach that accurately predicts short-term solutions and provides robust long-term statistics under diverse parameter conditions, facilitating computational cost savings and accelerating development in engineering simulations. We develop and compare parametric learning methods based on FNO and CNN, evaluating their effectiveness in learning parametric-dependent solution time-advancement operators for one-dimensional PDEs and realistic flame front evolution data obtained from direct numerical simulations of the Navier-Stokes equations.

An operator learning perspective on parameter-to-observable maps

Computationally efficient surrogates for parametrized physical models play a crucial role in science and engineering. Operator learning provides data-driven surrogates that map between function spaces. However, instead of full-field measurements, often the available data are only finite-dimensional parametrizations of model inputs or finite observables of model outputs. Building off of Fourier Neural Operators, this paper introduces the Fourier Neural Mappings (FNMs) framework that is able to accommodate such finite-dimensional inputs and outputs. The paper develops universal approximation theorems for the method. Moreover, in many applications the underlying parameter-to-observable (PtO) map is defined implicitly through an infinite-dimensional operator, such as the solution operator of a partial differential equation. A natural question is whether it is more data-efficient to learn the PtO map end-to-end or first learn the solution operator and subsequently compute the observable from the full-field solution. A theoretical analysis of Bayesian nonparametric regression of linear functionals, which is of independent interest, suggests that the end-to-end approach can actually have worse sample complexity. Extending beyond the theory, numerical results for the FNM approximation of three nonlinear PtO maps demonstrate the benefits of the operator learning perspective that this paper adopts.

Dynamic Gaussian Graph Operator: Learning parametric partial differential equations in arbitrary discrete mechanics problems

Deep learning methods have access to be employed for solving physical systems governed by parametric partial differential equations (PDEs) due to massive scientific data. It has been refined to operator learning that focuses on learning non-linear mapping between infinite-dimensional function spaces, offering interface from observations to solutions. However, state-of-the-art neural operators are limited to constant and uniform discretization, thereby leading to deficiency in generalization on arbitrary discretization schemes for computational domain. In this work, we propose a novel operator learning algorithm, referred to as Dynamic Gaussian Graph Operator (DGGO) that expands neural operators to learning parametric PDEs in arbitrary discrete mechanics problems. The Dynamic Gaussian Graph (DGG) kernel learns to map the observation vectors defined in general Euclidean space to metric vectors defined in high-dimensional uniform metric space. The DGG integral kernel is parameterized by Gaussian kernel weighted Riemann sum approximating and using dynamic message passing graph to depict the interrelation within the integral term. Fourier Neural Operator is selected to localize the metric vectors on spatial and frequency domains. Metric vectors are regarded as located on latent uniform domain, wherein spatial and spectral transformation offer highly regular constraints on solution space. The efficiency and robustness of DGGO are validated by applying it to solve numerical arbitrary discrete mechanics problems in comparison with mainstream neural operators. Ablation experiments are implemented to demonstrate the effectiveness of spatial transformation in the DGG kernel. The proposed method is utilized to forecast stress field of hyper-elastic material with geometrically variable void as engineering application.

GIT-Net: Generalized Integral Transform for Operator Learning

This article introduces GIT-Net, a deep neural network architecture for approximating Partial Differential Equation (PDE) operators, inspired by integral transform operators. GIT-NET harnesses the fact that differential operators commonly used for defining PDEs can often be represented parsimoniously when expressed in specialized functional bases (e.g., Fourier basis). Unlike rigid integral transforms, GIT-Net parametrizes adaptive generalized integral transforms with deep neural networks. When compared to several recently proposed alternatives, GIT-Net's computational and memory requirements scale gracefully with mesh discretizations, facilitating its application to PDE problems on complex geometries. Numerical experiments demonstrate that GIT-Net is a competitive neural network operator, exhibiting small test errors and low evaluations across a range of PDE problems. This stands in contrast to existing neural network operators, which typically excel in just one of these areas.