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Andrew Horning (Rensselaer Polytechnic Institute)5/19/26, 2:00 PMTheoretical Advances in Operator LearningMinisymposium Talk
Linear operators with a continuous spectrum often lurk behind complex physical phenomena in nature, from wave attractors in geometrically confined fluids to topological bifurcations in dynamical systems. However, they are notoriously tricky to learn from data. For example, finite-dimensional approximations of the operator must “discretize” the continuous spectrum into finitely many points and...
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Trevor Camper (Dartmouth College)5/19/26, 2:25 PMTheoretical Advances in Operator LearningMinisymposium Talk
Koopman operators and transfer operators represent nonlinear dynamics in state space through its induced action on linear spaces of observables and measures, respectively. This framework enables the use of linear operator theory for analysis and modeling of nonlinear dynamical systems, and has received considerable interest over the years from mathematical, computational, and domain-scientific...
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Bohan Chen (California Institute of Technology)5/19/26, 2:50 PMTheoretical Advances in Operator LearningMinisymposium Talk
Many classic methods in data assimilation, like the Ensemble Kalman Filter (EnKF), are limited by its Gaussian ansatz. In this work, we frame the filtering update as learning a nonlinear operator mapping between probability distributions in the mean-field limit. We introduce Measure Neural Mappings (MNMs), a class of neural operators acting on probability measures, implemented via Set...
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George Stepaniants (California Institute of Technology)5/19/26, 3:45 PMTheoretical Advances in Operator LearningMinisymposium Talk
The simulation of multiscale viscoelastic materials poses a significant challenge in computational materials science, requiring expensive numerical solvers that can resolve dynamics of material deformations at the microscopic scale. The theory of homogenization offers an alternative approach to modeling, by locally averaging the strains and stresses of multiscale materials. This procedure...
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Esther Gallmeier (Cornell University)5/19/26, 4:10 PMTheoretical Advances in Operator LearningMinisymposium Talk
The goal underlying our work is to develop provably accurate and data-efficient learning algorithms for non-self-adjoint operators using only input-output pairs. State-of-the-art approximation techniques with fast convergence rates either apply only to self-adjoint operators or require access to the adjoint operator, which is unavailable in experimental settings and often difficult to access...
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David Persson (New York University & Flatiron Institute)5/19/26, 4:35 PMTheoretical Advances in Operator LearningMinisymposium Talk
We present a randomized algorithm for producing a quasi-optimal hierarchically semi-separable (HSS) approximation to an $N\times N$ matrix $A$ using only matrix-vector products with $A$ and $A^T$. We prove that, using $O(k \log(N/k))$ matrix-vector products and $O(N k^2 \log(N/k))$ additional runtime, the algorithm returns an HSS matrix $B$ with rank-$k$ blocks whose expected Frobenius norm...
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Serkan Gugercin (Virginia Tech)5/19/26, 5:00 PMTheoretical Advances in Operator LearningMinisymposium Talk
We introduce a unified approach to $\mathcal{L}_2$-optimal reduced-order modeling that applies to both linear time-invariant dynamical systems and stationary parametric problems. The framework leverages parameter-separable representations to obtain gradient information for the $\mathcal{L}_2$ objective with respect to the reduced operators, enabling a fully nonintrusive, data-driven,...
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Aras Bacho (California Institute of Technology)5/21/26, 2:00 PMTheoretical Advances in Operator LearningMinisymposium Talk
Neural operator learning methods have garnered significant attention in scientific computing for their ability to approximate infinite-dimensional operators. However, increasing their complexity often fails to substantially improve their accuracy, leaving them on par with much simpler approaches such as kernel methods and more traditional reduced-order models. In this article, we set out to...
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Juan Felipe Osorio Ramirez (University of Washington)5/21/26, 2:25 PMTheoretical Advances in Operator LearningMinisymposium Talk
We present an alternative perspective on operator learning for problems in which the operator is implicitly defined by a partial differential equation. Rather than learning the solution operator directly as a high-dimensional mapping, we propose to first learn the underlying PDE operator as a local differential operator and then numerically invert it to evaluate the associated solution operator.
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Gil Goldshlager (UC Berkeley)5/21/26, 2:50 PMTheoretical Advances in Operator LearningMinisymposium Talk
An increasing number of theoretical results are available to characterize the extent to which neural networks can (i) represent scientifically relevant functions and operators, and (ii) learn these functions and operators from data. However, even with the right network architecture and the right dataset, optimization is a bottleneck. On the one hand, popular machine learning optimizers such as...
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Christopher Beattie (Virginia Tech)5/21/26, 3:15 PMTheoretical Advances in Operator LearningMinisymposium Talk
Gaussian processes (GPs) defined through intrinsic random fields provide a flexible framework for modeling spatial phenomena, and have been advocated in a variety of applications over the past several decades. Nevertheless, their adoption has lagged behind traditional, covariance-based approaches, in part because the intrinsic formulation has lacked an accompanying toolkit of computational...
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