Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs.

This paper establishes convergence rates for learning elliptic pseudo-differential operators, a fundamental operator class in partial differential equations and mathematical physics. In a wavelet-Galerkin framework, we formulate learning over this class as a structured infinite-dimensional regression problem with multiscale sparsity. Building on this structure, we propose a sparse, data- and computation-efficient estimator, which leverages a novel matrix compression scheme tailored to the learning task and a nested-support strategy to balance approximation and estimation errors. In addition to obtaining convergence rates for the estimator, we show that the learned operator induces an efficient and stable Galerkin solver whose numerical error matches its statistical accuracy. Our results therefore contribute to bringing together operator learning, data-driven solvers, and wavelet methods in scientific computing.

Machine learning approaches for solving partial differential equations require learning mappings between function spaces. While convolutional or graph neural networks are constrained to discretized functions, neural operators present a promising milestone toward mapping functions directly. Despite impressive results they still face challenges with respect to the domain geometry and typically rely on some form of discretization. In order to alleviate such limitations, we present CORAL, a new method that leverages coordinate-based networks for solving PDEs on general geometries. CORAL is designed to remove constraints on the input mesh, making it applicable to any spatial sampling and geometry. Its ability extends to diverse problem domains, including PDE solving, spatio-temporal forecasting, and inverse problems like geometric design. CORAL demonstrates robust performance across multiple resolutions and performs well in both convex and non-convex domains, surpassing or performing on par with state-of-the-art models.

Supervised learning in function spaces is an emerging area of machine learning research with applications to the prediction of complex physical systems such as fluid flows, solid mechanics, and climate modeling. By directly learning maps (operators) between infinite dimensional function spaces, these models are able to learn discretization invariant representations of target functions. A common approach is to represent such target functions as linear combinations of basis elements learned from data. However, there are simple scenarios where, even though the target functions form a low dimensional submanifold, a very large number of basis elements is needed for an accurate linear representation.

A new paradigm in machine learning for scientific computing is focused on designing learning algorithms and methods for continuum problems. This paradigm is referred to as operator learning and has received considerable interest in the last few years [5,7,18,20,23-25,27,30,34,36]. The basic task may be posed as learning a map between infinite-dimensional function spaces, i.e., learning an operator F: X Y, where, for example, X and Y are real, separable Hilbert spaces. Operator learning naturally arises in many scientific problems where one wants to learn how a continuum model, often described by partial differential equations (PDEs), maps inputs, such as parameters or boundary conditions, to outputs, such as states or observables. A prototypical example to keep in mind is learning parameter-to-solution maps of parametric PDEs [1,2,11]. In contrast to more classical surrogate modeling, which typically focuses on learning finite-dimensional parameter-to-solution maps for some fixed discretization, operator learning directly aims to learn/approximate the continuum map F: X Y itself. Thus, the inputs and outputs are functions (not vectors) and the goal is to directly design discretization-invariant methods [7,23]. From a statistical perspective, this naturally leads to a nonparametric regression problem in which both the object of interest (the operator) and the observations (finite number of noisy samples) are infinite-dimensional.

Deep Operator Networks (DeepONets) have become a central tool in data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk design based on fully connected layers acting on raw spatial or spatiotemporal coordinates struggles to represent sharp gradients, boundary layers, and non-periodic structures commonly found in PDEs posed on bounded domains with Dirichlet or Neumann boundary conditions. To address these limitations, we introduce the Spectral-Embedded DeepONet (SEDONet), a new DeepONet variant in which the trunk is driven by a fixed Chebyshev spectral dictionary rather than coordinate inputs. This non-periodic spectral embedding provides a principled inductive bias tailored to bounded domains, enabling the learned operator to capture fine-scale non-periodic features that are difficult for Fourier or MLP trunks to represent. SEDONet is evaluated on a suite of PDE benchmarks including 2D Poisson, 1D Burgers, 1D advection-diffusion, Allen-Cahn dynamics, and the Lorenz-96 chaotic system, covering elliptic, parabolic, advective, and multiscale temporal phenomena, all of which can be viewed as canonical problems in computational mechanics. Across all datasets, SEDONet consistently achieves the lowest relative L2 errors among DeepONet, FEDONet, and SEDONet, with average improvements of about 30-40% over the baseline DeepONet and meaningful gains over Fourier-embedded variants on non-periodic geometries. Spectral analyses further show that SEDONet more accurately preserves high-frequency and boundary-localized features, demonstrating the value of Chebyshev embeddings in non-periodic operator learning. The proposed architecture offers a simple, parameter-neutral modification to DeepONets, delivering a robust and efficient spectral framework for surrogate modeling of PDEs on bounded domains.

Operator learning offers a robust framework for approximating mappings between infinite-dimensional function spaces. It has also become a powerful tool for solving inverse problems in the computational sciences. This chapter surveys methodological and theoretical developments at the intersection of operator learning and inverse problems. It begins by summarizing the probabilistic and deterministic approaches to inverse problems, and pays special attention to emerging measure-centric formulations that treat observed data or unknown parameters as probability distributions. The discussion then turns to operator learning by covering essential components such as data generation, loss functions, and widely used architectures for representing function-to-function maps. The core of the chapter centers on the end-to-end inverse operator learning paradigm, which aims to directly map observed data to the solution of the inverse problem without requiring explicit knowledge of the forward map. It highlights the unique challenge that noise plays in this data-driven inversion setting, presents structure-aware architectures for both point predictions and posterior estimates, and surveys relevant theory for linear and nonlinear inverse problems. The chapter also discusses the estimation of priors and regularizers, where operator learning is used more selectively within classical inversion algorithms.

Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs.

Industrial design evaluation often relies on high-fidelity simulations of governing partial differential equations (PDEs). While accurate, these simulations are computationally expensive, making dense exploration of design spaces impractical. Operator learning has emerged as a promising approach to accelerate PDE solution prediction; however, its effectiveness is often limited by the scarcity of labeled physics-based data. At the same time, large numbers of geometry-only candidate designs are readily available but remain largely untapped. We propose a two-stage framework to better exploit this abundant, physics-agnostic resource and improve supervised operator learning under limited labeled data. In Stage 1, we pretrain an autoencoder on a geometry reconstruction task to learn an expressive latent representation without PDE labels. In Stage 2, the neural operator is trained in a standard supervised manner to predict PDE solutions, using the pretrained latent embeddings as inputs instead of raw point clouds. Transformer-based architectures are adopted for both the autoencoder and the neural operator to handle point cloud data and integrate both stages seamlessly. Across four PDE datasets and three state-of-the-art transformer-based neural operators, our approach consistently improves prediction accuracy compared to models trained directly on raw point cloud inputs. These results demonstrate that representations from physics-agnostic pretraining provide a powerful foundation for data-efficient operator learning.

Accurate modeling of personalized cardiovascular dynamics is crucial for non-invasive monitoring and therapy planning. State-of-the-art physics-informed neural network (PINN) approaches employ deep, multi-branch architectures with adversarial or contrastive objectives to enforce partial differential equation constraints. While effective, these enhancements introduce significant training and implementation complexity, limiting scalability and practical deployment. We investigate physics-informed neural operator learning models as efficient supervisory signals for training simplified architectures through knowledge distillation. Our approach pre-trains a physics-informed DeepONet (PI-DeepONet) on high-fidelity cuffless blood pressure recordings to learn operator mappings from raw wearable waveforms to beat-to-beat pressure signals under embedded physics constraints. This pre-trained operator serves as a frozen supervisor in a lightweight knowledge-distillation pipeline, guiding streamlined base models that eliminate complex adversarial and contrastive learning components while maintaining performance. We characterize the role of physics-informed regularization in operator learning and demonstrate its effectiveness for supervisory guidance. Through extensive experiments, our operator-supervised approach achieves performance parity with complex baselines (correlation: 0.766 vs. 0.770, RMSE: 4.452 vs. 4.501), while dramatically reducing architectural complexity from eight critical hyperparameters to a single regularization coefficient and decreasing training overhead by 4%. Our results demonstrate that operator-based supervision effectively replaces intricate multi-component training strategies, offering a more scalable and interpretable approach to physiological modeling with reduced implementation burden.