Operator learning, which aims to approximate maps between infinite-dimensional function spaces, is an important area in scientific machine learning with applications across various physical domains. Here we introduce the Continuous Vision Transformer (CViT), a novel neural operator architecture that leverages advances in computer vision to address challenges in learning complex physical systems. CViT combines a vision transformer encoder, a novel grid-based coordinate embedding, and a query-wise cross-attention mechanism to effectively capture multi-scale dependencies. This design allows for flexible output representations and consistent evaluation at arbitrary resolutions. We demonstrate CViT's effectiveness across a diverse range of partial differential equation (PDE) systems, including fluid dynamics, climate modeling, and reaction-diffusion processes. Our comprehensive experiments show that CViT achieves state-of-the-art performance on multiple benchmarks, often surpassing larger foundation models, even without extensive pretraining and roll-out fine-tuning. Taken together, CViT exhibits robust handling of discontinuous solutions, multi-scale features, and intricate spatio-temporal dynamics. Our contributions can be viewed as a significant step towards adapting advanced computer vision architectures for building more flexible and accurate machine learning models in the physical sciences.
The joint prediction of continuous fields and statistical estimation of the underlying discrete parameters is a common problem for many physical systems, governed by PDEs. Hitherto, it has been separately addressed by employing operator learning surrogates for field prediction while using simulation-based inference (and its variants) for statistical parameter determination. Here, we argue that solving both problems within the same framework can lead to consistent gains in accuracy and robustness. To this end, We propose a novel and flexible formulation of the operator learning problem that allows jointly predicting continuous quantities and inferring distributions of discrete parameters, and thus amortizing the cost of both the inverse and the surrogate models to a joint pre-training step. We present the capabilities of the proposed methodology for predicting continuous and discrete biomarkers in full-body haemodynamics simulations under different levels of missing information. We also consider a test case for atmospheric large-eddy simulation of a two-dimensional dry cold bubble, where we infer both continuous time-series and information about the systems conditions. We present comparisons against different baselines to showcase significantly increased accuracy in both the inverse and the surrogate tasks.
We propose a novel finite element-based physics-informed operator learning framework that allows for predicting spatiotemporal dynamics governed by partial differential equations (PDEs). The proposed framework employs a loss function inspired by the finite element method (FEM) with the implicit Euler time integration scheme. A transient thermal conduction problem is considered to benchmark the performance. The proposed operator learning framework takes a temperature field at the current time step as input and predicts a temperature field at the next time step. The Galerkin discretized weak formulation of the heat equation is employed to incorporate physics into the loss function, which is coined finite operator learning (FOL). Upon training, the networks successfully predict the temperature evolution over time for any initial temperature field at high accuracy compared to the FEM solution. The framework is also confirmed to be applicable to a heterogeneous thermal conductivity and arbitrary geometry. The advantages of FOL can be summarized as follows: First, the training is performed in an unsupervised manner, avoiding the need for a large data set prepared from costly simulations or experiments. Instead, random temperature patterns generated by the Gaussian random process and the Fourier series, combined with constant temperature fields, are used as training data to cover possible temperature cases. Second, shape functions and backward difference approximation are exploited for the domain discretization, resulting in a purely algebraic equation. This enhances training efficiency, as one avoids time-consuming automatic differentiation when optimizing weights and biases while accepting possible discretization errors. Finally, thanks to the interpolation power of FEM, any arbitrary geometry can be handled with FOL, which is crucial to addressing various engineering application scenarios.
We present a novel deep operator network (DeepONet) architecture for operator learning, the ensemble DeepONet, that allows for enriching the trunk network of a single DeepONet with multiple distinct trunk networks. This trunk enrichment allows for greater expressivity and generalization capabilities over a range of operator learning problems. We also present a spatial mixture-of-experts (MoE) DeepONet trunk network architecture that utilizes a partition-of-unity (PoU) approximation to promote spatial locality and model sparsity in the operator learning problem. We first prove that both the ensemble and PoU-MoE DeepONets are universal approximators. We then demonstrate that ensemble DeepONets containing a trunk ensemble of a standard trunk, the PoU-MoE trunk, and/or a proper orthogonal decomposition (POD) trunk can achieve 2-4x lower relative $\ell_2$ errors than standard DeepONets and POD-DeepONets on both standard and challenging new operator learning problems involving partial differential equations (PDEs) in two and three dimensions. Our new PoU-MoE formulation provides a natural way to incorporate spatial locality and model sparsity into any neural network architecture, while our new ensemble DeepONet provides a powerful and general framework for incorporating basis enrichment in scientific machine learning architectures for operator learning.
In this work, we present an arbitrary-scale super-resolution (SR) method to enhance the resolution of scientific data, which often involves complex challenges such as continuity, multi-scale physics, and the intricacies of high-frequency signals. Grounded in operator learning, the proposed method is resolution-invariant. The core of our model is a hierarchical neural operator that leverages a Galerkin-type self-attention mechanism, enabling efficient learning of mappings between function spaces. Sinc filters are used to facilitate the information transfer across different levels in the hierarchy, thereby ensuring representation equivalence in the proposed neural operator. Additionally, we introduce a learnable prior structure that is derived from the spectral resizing of the input data. This loss prior is model-agnostic and is designed to dynamically adjust the weighting of pixel contributions, thereby balancing gradients effectively across the model. We conduct extensive experiments on diverse datasets from different domains and demonstrate consistent improvements compared to strong baselines, which consist of various state-of-the-art SR methods.
Operator learning for Partial Differential Equations (PDEs) is rapidly emerging as a promising approach for surrogate modeling of intricate systems. Transformers with the self-attention mechanism$\unicode{x2013}$a powerful tool originally designed for natural language processing$\unicode{x2013}$have recently been adapted for operator learning. However, they confront challenges, including high computational demands and limited interpretability. This raises a critical question: Is there a more efficient attention mechanism for Transformer-based operator learning? This paper proposes the Position-induced Transformer (PiT), built on an innovative position-attention mechanism, which demonstrates significant advantages over the classical self-attention in operator learning. Position-attention draws inspiration from numerical methods for PDEs. Different from self-attention, position-attention is induced by only the spatial interrelations of sampling positions for input functions of the operators, and does not rely on the input function values themselves, thereby greatly boosting efficiency. PiT exhibits superior performance over current state-of-the-art neural operators in a variety of complex operator learning tasks across diverse PDE benchmarks. Additionally, PiT possesses an enhanced discretization convergence feature, compared to the widely-used Fourier neural operator.
The Koopman operator provides a powerful framework for representing the dynamics of general nonlinear dynamical systems. Data-driven techniques to learn the Koopman operator typically assume that the chosen function space is closed under system dynamics. In this paper, we study the Koopman operator via its action on the reproducing kernel Hilbert space (RKHS), and explore the mis-specified scenario where the dynamics may escape the chosen function space. We relate the Koopman operator to the conditional mean embeddings (CME) operator and then present an operator stochastic approximation algorithm to learn the Koopman operator iteratively with control over the complexity of the representation. We provide both asymptotic and finite-time last-iterate guarantees of the online sparse learning algorithm with trajectory-based sampling with an analysis that is substantially more involved than that for finite-dimensional stochastic approximation. Numerical examples confirm the effectiveness of the proposed algorithm.
This paper presents a novel Koopman (composition) operator representation framework for control systems in reproducing kernel Hilbert spaces (RKHSs) that is free of explicit dictionary or input parametrizations. By establishing fundamental equivalences between different model representations, we are able to close the gap of control system operator learning and infinite-dimensional regression, enabling various empirical estimators and the connection to well-understood learning theory in RKHSs under one unified framework. As a consequence, our proposed framework allows for arbitrary accurate finite-rank approximations in infinite-dimensional spaces and leads to finite-dimensional predictors without apriori restrictions to a finite span of functions or inputs. To enable applications to high-dimensional control systems, we improve the scalability of our proposed control Koopman operator estimates by utilizing sketching techniques. Numerical experiments demonstrate superior prediction accuracy compared to bilinear EDMD, especially in high dimensions. Finally, we show that our learned models are readily interfaced with linear-parameter-varying techniques for model predictive control.
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.
Operator learning is a variant of machine learning that is designed to approximate maps between function spaces from data. The Fourier Neural Operator (FNO) is a common model architecture used for operator learning. The FNO combines pointwise linear and nonlinear operations in physical space with pointwise linear operations in Fourier space, leading to a parameterized map acting between function spaces. Although FNOs formally involve convolutions of functions on a continuum, in practice the computations are performed on a discretized grid, allowing efficient implementation via the FFT. In this paper, the aliasing error that results from such a discretization is quantified and algebraic rates of convergence in terms of the grid resolution are obtained as a function of the regularity of the input. Numerical experiments that validate the theory and describe model stability are performed.
In this study, we develop a novel multi-fidelity deep learning approach that transforms low-fidelity solution maps into high-fidelity ones by incorporating parametric space information into a standard autoencoder architecture. This method's integration of parametric space information significantly reduces the need for training data to effectively predict high-fidelity solutions from low-fidelity ones. In this study, we examine a two-dimensional steady-state heat transfer analysis within a highly heterogeneous materials microstructure. The heat conductivity coefficients for two different materials are condensed from a 101 x 101 grid to smaller grids. We then solve the boundary value problem on the coarsest grid using a pre-trained physics-informed neural operator network known as Finite Operator Learning (FOL). The resulting low-fidelity solution is subsequently upscaled back to a 101 x 101 grid using a newly designed enhanced autoencoder. The novelty of the developed enhanced autoencoder lies in the concatenation of heat conductivity maps of different resolutions to the decoder segment in distinct steps. Hence the developed algorithm is named microstructure-embedded autoencoder (MEA). We compare the MEA outcomes with those from finite element methods, the standard U-Net, and various other upscaling techniques, including interpolation functions and feedforward neural networks (FFNN). Our analysis shows that MEA outperforms these methods in terms of computational efficiency and error on test cases. As a result, the MEA serves as a potential supplement to neural operator networks, effectively upscaling low-fidelity solutions to high fidelity while preserving critical details often lost in traditional upscaling methods, particularly at sharp interfaces like those seen with interpolation.
In this paper, we provide a theoretical analysis of a type of operator learning method without data reliance based on the classical finite element approximation, which is called the finite element operator network (FEONet). We first establish the convergence of this method for general second-order linear elliptic PDEs with respect to the parameters for neural network approximation. In this regard, we address the role of the condition number of the finite element matrix in the convergence of the method. Secondly, we derive an explicit error estimate for the self-adjoint case. For this, we investigate some regularity properties of the solution in certain function classes for a neural network approximation, verifying the sufficient condition for the solution to have the desired regularity. Finally, we will also conduct some numerical experiments that support the theoretical findings, confirming the role of the condition number of the finite element matrix in the overall convergence.
We propose a new neural network based method for solving inverse problems for partial differential equations (PDEs) by formulating the PDE inverse problem as a bilevel optimization problem. At the upper level, we minimize the data loss with respect to the PDE parameters. At the lower level, we train a neural network to locally approximate the PDE solution operator in the neighborhood of a given set of PDE parameters, which enables an accurate approximation of the descent direction for the upper level optimization problem. The lower level loss function includes the L2 norms of both the residual and its derivative with respect to the PDE parameters. We apply gradient descent simultaneously on both the upper and lower level optimization problems, leading to an effective and fast algorithm. The method, which we refer to as BiLO (Bilevel Local Operator learning), is also able to efficiently infer unknown functions in the PDEs through the introduction of an auxiliary variable. Through extensive experiments over multiple PDE systems, we demonstrate that our method enforces strong PDE constraints, is robust to sparse and noisy data, and eliminates the need to balance the residual and the data loss, which is inherent to the soft PDE constraints in many existing methods.
We propose a neural operator framework, termed mixture density nonlinear manifold decoder (MD-NOMAD), for stochastic simulators. Our approach leverages an amalgamation of the pointwise operator learning neural architecture nonlinear manifold decoder (NOMAD) with mixture density-based methods to estimate conditional probability distributions for stochastic output functions. MD-NOMAD harnesses the ability of probabilistic mixture models to estimate complex probability and the high-dimensional scalability of pointwise neural operator NOMAD. We conduct empirical assessments on a wide array of stochastic ordinary and partial differential equations and present the corresponding results, which highlight the performance of the proposed framework.
The study of neural operators has paved the way for the development of efficient approaches for solving partial differential equations (PDEs) compared with traditional methods. However, most of the existing neural operators lack the capability to provide uncertainty measures for their predictions, a crucial aspect, especially in data-driven scenarios with limited available data. In this work, we propose a novel Neural Operator-induced Gaussian Process (NOGaP), which exploits the probabilistic characteristics of Gaussian Processes (GPs) while leveraging the learning prowess of operator learning. The proposed framework leads to improved prediction accuracy and offers a quantifiable measure of uncertainty. The proposed framework is extensively evaluated through experiments on various PDE examples, including Burger's equation, Darcy flow, non-homogeneous Poisson, and wave-advection equations. Furthermore, a comparative study with state-of-the-art operator learning algorithms is presented to highlight the advantages of NOGaP. The results demonstrate superior accuracy and expected uncertainty characteristics, suggesting the promising potential of the proposed framework.
The Kernel-Free Boundary Integral (KFBI) method presents an iterative solution to boundary integral equations arising from elliptic partial differential equations (PDEs). This method effectively addresses elliptic PDEs on irregular domains, including the modified Helmholtz, Stokes, and elasticity equations. The rapid evolution of neural networks and deep learning has invigorated the exploration of numerical PDEs. An increasing interest is observed in deep learning approaches that seamlessly integrate mathematical principles for investigating numerical PDEs. We propose a hybrid KFBI method, integrating the foundational principles of the KFBI method with the capabilities of deep learning. This approach, within the framework of the boundary integral method, designs a network to approximate the solution operator for the corresponding integral equations by mapping the parameters, inhomogeneous terms and boundary information of PDEs to the boundary density functions, which can be regarded as the solution of the integral equations. The models are trained using data generated by the Cartesian grid-based KFBI algorithm, exhibiting robust generalization capabilities. It accurately predicts density functions across diverse boundary conditions and parameters within the same class of equations. Experimental results demonstrate that the trained model can directly infer the boundary density function with satisfactory precision, obviating the need for iterative steps in solving boundary integral equations. Furthermore, applying the inference results of the model as initial values for iterations is also reasonable; this approach can retain the inherent second-order accuracy of the KFBI method while accelerating the traditional KFBI approach by reducing about 50% iterations.
Foundation models, such as large language models, have demonstrated success in addressing various language and image processing tasks. In this work, we introduce a multi-modal foundation model for scientific problems, named PROSE-PDE. Our model, designed for bi-modality to bi-modality learning, is a multi-operator learning approach which can predict future states of spatiotemporal systems while concurrently learning the underlying governing equations of the physical system. Specifically, we focus on multi-operator learning by training distinct one-dimensional time-dependent nonlinear constant coefficient partial differential equations, with potential applications to many physical applications including physics, geology, and biology. More importantly, we provide three extrapolation studies to demonstrate that PROSE-PDE can generalize physical features through the robust training of multiple operators and that the proposed model can extrapolate to predict PDE solutions whose models or data were unseen during the training. Furthermore, we show through systematic numerical experiments that the utilization of the symbolic modality in our model effectively resolves the well-posedness problems with training multiple operators and thus enhances our model's predictive capabilities.
Since infectious pathogens start spreading into a susceptible population, mathematical models can provide policy makers with reliable forecasts and scenario analyses, which can be concretely implemented or solely consulted. In these complex epidemiological scenarios, machine learning architectures can play an important role, since they directly reconstruct data-driven models circumventing the specific modelling choices and the parameter calibration, typical of classical compartmental models. In this work, we discuss the efficacy of Kernel Operator Learning (KOL) to reconstruct population dynamics during epidemic outbreaks, where the transmission rate is ruled by an input strategy. In particular, we introduce two surrogate models, named KOL-m and KOL-$\partial$, which reconstruct in two different ways the evolution of the epidemics. Moreover, we evaluate the generalization performances of the two approaches with different kernels, including the Neural Tangent Kernels, and compare them with a classical neural network model learning method. Employing synthetic but semi-realistic data, we show how the two introduced approaches are suitable for realizing fast and robust forecasts and scenario analyses, and how these approaches are competitive for determining optimal intervention strategies with respect to specific performance measures.
In this paper, we construct a mixture of neural operators (MoNOs) between function spaces whose complexity is distributed over a network of expert neural operators (NOs), with each NO satisfying parameter scaling restrictions. Our main result is a \textit{distributed} universal approximation theorem guaranteeing that any Lipschitz non-linear operator between $L^2([0,1]^d)$ spaces can be approximated uniformly over the Sobolev unit ball therein, to any given $\varepsilon>0$ accuracy, by an MoNO while satisfying the constraint that: each expert NO has a depth, width, and rank of $\mathcal{O}(\varepsilon^{-1})$. Naturally, our result implies that the required number of experts must be large, however, each NO is guaranteed to be small enough to be loadable into the active memory of most computers for reasonable accuracies $\varepsilon$. During our analysis, we also obtain new quantitative expression rates for classical NOs approximating uniformly continuous non-linear operators uniformly on compact subsets of $L^2([0,1]^d)$.
Homogenization is a fundamental tool for studying multiscale physical phenomena. Traditional numerical homogenization methods, heavily reliant on finite element analysis, demand significant computational resources, especially for complex geometries, materials, and high-resolution problems. To address these challenges, we propose PreFine-Homo, a novel numerical homogenization framework comprising two phases: pretraining and fine-tuning. In the pretraining phase, a Fourier Neural Operator (FNO) is trained on large datasets to learn the mapping from input geometries and material properties to displacement fields. In the fine-tuning phase, the pretrained predictions serve as initial solutions for iterative algorithms, drastically reducing the number of iterations needed for convergence. The pretraining phase of PreFine-Homo delivers homogenization results up to 1000 times faster than conventional methods, while the fine-tuning phase further enhances accuracy. Moreover, the fine-tuning phase grants PreFine-Homo unlimited generalization capabilities, enabling continuous learning and improvement as data availability increases. We validate PreFine-Homo by predicting the effective elastic tensor for 3D periodic materials, specifically Triply Periodic Minimal Surfaces (TPMS). The results demonstrate that PreFine-Homo achieves high precision, exceptional efficiency, robust learning capabilities, and strong extrapolation ability, establishing it as a powerful tool for multiscale homogenization tasks.