Operator Learning

Despite the successes of deep learning in computer vision, difficulties persist in recognizing objects that have undergone group-symmetric transformations rarely seen during training$\unicode{x2013}$for example objects seen in unusual poses, scales, positions, or combinations thereof. Equivariant neural networks are a solution to the problem of generalizing across symmetric transformations, but require knowledge of transformations a priori. An alternative family of architectures proposes to learn equivariant operators in a latent space, from examples of symmetric transformations. Here, using simple datasets of rotated and translated noisy MNIST, we illustrate how such architectures can successfully be harnessed for out-of-distribution classification, thus overcoming the limitations of both traditional and equivariant networks. While conceptually enticing, we discuss challenges ahead on the path of scaling these architectures to more complex datasets. Our code is available at https://github.com/BRAIN-Aalto/equivariant_operator.

We present a novel property-preserving kernel-based operator learning method for incompressible flows governed by the incompressible Navier--Stokes equations. Traditional numerical solvers incur significant computational costs to respect incompressibility. Operator learning offers efficient surrogate models, but current neural operators fail to exactly enforce physical properties such as incompressibility, periodicity, and turbulence. Our kernel method maps input functions to expansion coefficients of output functions in a property-preserving kernel basis, ensuring that predicted velocity fields analytically and simultaneously preserve the aforementioned physical properties. We present universal approximation results and worst-case a priori convergence rates for our framework; empirically, the observed convergence rates exceed the pessimistic predictions across most benchmarks, motivating a formulation of more optimistic convergence rates. Another central contribution of this work is a novel computational framework revolving around streaming construction of kernel Gramians and recursive Schur-complement algorithms to solve the large block linear systems arising from property-preserving kernel methods. This kernel-based framework makes operator learning practical at scales up to $10{,}000$ training functions sampled at up to $10{,}000$ spatial locations. We evaluate the method on challenging 2D and 3D, laminar and turbulent, incompressible flow problems. Our method achieves up to six orders of magnitude lower relative $\ell_2$ errors upon generalization and trains up to five orders of magnitude faster compared to neural operators. Moreover, while our method enforces incompressibility analytically, neural operators exhibit large divergence errors. Our results show that our method provides an accurate and efficient surrogate for incompressible flows.

An important application of neural networks in scientific computing is non-linear operator learning. In this framework, a neural network is trained to fit a non-linear map between infinite dimensional spaces, for example, the solution operator of ordinary and partial differential equations. Recently, inspired by the discovery of in-context learning for large language models, an even more ambitious paradigm has been explored, called multi-operator learning. In this approach, a neural network is trained to learn many different operators at the same time. At inference time, the network is passed a small number of example input and output pairs to disambiguate the desired operator. In this work, we provide a precise mathematical formulation of the multi-operator learning problem, propose a parameter-efficient, attention-free architecture, called DeepOSets, and prove its universality in the in-context multi-operator learning task. We perform experiments demonstrating the efficacy of DeepOSets on several benchmarks, including Burgers' equation, 1D and 2D Darcy flow, and a variety of initial-value and boundary-value differential equations. In a comparison with the transformer-based ICON model on the 1D Darcy problem, under a small training budget (10k steps), DeepOSets attained lower error at every context size with 19% of the parameters, while with an extended budget (100k steps), ICON attains 26-41% lower error, but at the cost of roughly 218$\times$ more training time, 12$\times$ higher inference latency, and 5.3$\times$ more parameters.

Learned image reconstruction has become a pillar in computational imaging and inverse problems. Among the most successful approaches are learned iterative networks, which are formulated by unrolling classical iterative optimisation algorithms for solving variational problems. While the underlying algorithm is usually formulated in the functional analytic setting, learned approaches are often viewed as purely discrete. In this survey we present a unified operator view for learned iterative networks. Specifically, we formulate a learned reconstruction operator, defining how to compute, and separately the learning problem, which defines what to compute. In this setting we present common approaches and show that many approaches are closely related in their core. We review linear as well as non-linear inverse problems in this framework and present a short numerical study to conclude.

Continuous-time stochastic processes underlie many natural and engineered systems. In healthcare, autonomous driving, and industrial control, direct interaction with the environment is often unsafe or impractical, motivating offline reinforcement learning from historical data. However, there is limited statistical understanding of the approximation errors inherent in learning policies from offline datasets. We address this by linking reinforcement learning to the Hamilton-Jacobi-Bellman equation and proposing an operator-theoretic algorithm based on a simple dynamic programming recursion. Specifically, we represent our world model in terms of the infinitesimal generator of controlled diffusion processes learned in a reproducing kernel Hilbert space. By integrating statistical learning methods and operator theory, we establish global convergence of the value function under sufficiently small generator-approximation error and derive explicit finite-sample guarantees with bounds tied directly to system properties such as smoothness and stability. Our theoretical and proof-of-concept results indicate that operator-based approaches may hold promise for solving offline reinforcement learning using continuous-time optimal control.

We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) operator-valued kernels whose associated integral operators are compact and hence admit discrete spectral decompositions, and (ii) separable kernels of the form $K(x,x')=k(x,x')T$, where $k$ is a scalar-valued kernel and $T$ is a positive operator on the output space. This broad setting induces expressive vector-valued reproducing kernel Hilbert spaces (RKHSs) that generalize the classical $K=kI$ paradigm, thereby enabling rich structural modeling with rigorous theoretical guarantees. To address target operators lying outside the RKHS, we introduce vector-valued interpolation spaces to precisely quantify misspecification error. Within this framework, we establish non-asymptotic convergence rates for prediction, estimation, and misspecification errors in the online and finite-horizon settings. Importantly, the framework also accommodates a range of operator learning settings, from Fredholm integral operators to encoder--decoder architectures. Numerical experiments on the two-dimensional Navier--Stokes equations illustrate the proposed approach.

Neural operators (NOs) are a class of deep learning models designed to simultaneously solve infinitely many related problems by casting them into an infinite-dimensional space, whereon these NOs operate. Such operator learning problems arise naturally in mathematical finance, for instance, when learning families of optimal portfolios or hedging strategies. Two significant gaps, however, remain between theory and practice: 1) most universal approximation guarantees do not describe practically implementable NOs or are pure existence statements; and 2) worst-case parameter bounds from universal approximation theorems suggest that constructed NOs may require an unrealistically large number of parameters to solve most operator learning problems, which stands in direct opposition to a slew of experimental evidence. This paper closes these gaps by exhibiting a short and simple training algorithm to construct a specific class of NOs, proximal equilibrium operators (PEOs), using (realistic) finite-dimensional deep equilibrium layers to solve families of convex optimization problems over a separable Hilbert space $X$. Here, the inputs are smooth, convex loss functions on $X$, and the outputs are the associated (approximate) solutions to the optimization problems defined by each input loss. Our main result is a certificate showing that, when the input losses lie in a suitable infinite-dimensional set, our PEOs trained with our algorithm do in fact uniformly approximate the corresponding solutions to arbitrary precision, with rank and width growing only logarithmically and the depth growing linearly in the reciprocal of the approximation error. We then validate both our theoretical results and the trainability of PEOs on three applications: non-linear PDEs, stochastic optimal control, and portfolio optimization and hedging problems in mathematical finance.

We develop a new and general encode-approximate-reconstruct operator learning model that leverages learned neural representations of bases for input and output function distributions. We introduce the concepts of numerical operator learning and discretization independence, which clarify the relationship between theoretical formulations and practical realizations of operator learning models. Our model is discretization-independent, making it particularly effective for multiresolution learning. We establish theoretical approximation guarantees, demonstrating uniform universal approximation under strong assumptions on the input functions and statistical approximation under weaker conditions. To our knowledge, this is the first comprehensive study that investigates how discretization independence enables robust and efficient multiresolution operator learning. We validate our method through extensive numerical experiments involving both local and nonlocal PDEs, including time-independent and time-dependent problems. The results show that multiresolution training significantly improves accuracy and computational efficiency. Moreover, multiresolution training further enhances empirical discretization independence.

Motivated by the rapidly growing field of mathematics for operator approximation with neural networks, we present a novel universal operator approximation theorem for broad classes of encoder-decoder architectures and a wide range of input and output spaces. In this study, we focus on the approximation of continuous operators between infinite-dimensional normed or metric spaces in the topology of uniform convergence on compact sets. Unlike standard results in the operator learning literature, we additionally investigate the case where the approximating sequence of encoder-decoder architectures can be chosen independently of the compact sets. Taking a topological perspective, we point out that compact-set-independent approximation is a strictly stronger property in most relevant operator learning frameworks. To establish our results, we introduce new approximation properties of input and output spaces tailored to encoder-decoder architectures. These properties enable us to prove a universal operator approximation theorem ensuring uniform convergence on every compact subset of the input space. Our results unify and extend existing universal operator approximation theorems for various encoder-decoder architectures, including classical DeepONets, BasisONets, MIONets, architectures based on frames and other related approaches. A notable feature of our framework is that it also applies to metric spaces beyond the normed setting. In particular, it allows the consideration of $p$-Wasserstein spaces of probability measures as input or output spaces, and Skorohod spaces of càdlàg functions as input spaces. This generality also opens up potential applications in optimal transport.