Related papers: Fast operator learning for mapping correlations
The inductive biases of trained neural networks are difficult to understand and, consequently, to adapt to new settings. We study the inductive biases of linearizations of neural networks, which we show to be surprisingly good summaries of…
In this paper, we apply the self-attention from the state-of-the-art Transformer in Attention Is All You Need for the first time to a data-driven operator learning problem related to partial differential equations. An effort is put together…
This work studies reduced order modeling (ROM) approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models. It is shown that a key requirement for a successful reduced order…
We employ the framework of the Koopman operator and dynamic mode decomposition to devise a computationally cheap and easily implementable method to detect transient dynamics and regime changes in time series. We argue that typically…
We investigate reduced-order models for acoustic and electromagnetic wave problems in parametrically defined domains. The parameter-to-solution maps are approximated following the so-called Galerkin POD-NN method, which combines the…
A classical reduced order model (ROM) for dynamical problems typically involves only the spatial reduction of a given problem. Recently, a novel space-time ROM for linear dynamical problems has been developed, which further reduces the…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
This paper presents fast non-sampling based methods to assess the risk of trajectories for autonomous vehicles when probabilistic predictions of other agents' futures are generated by deep neural networks (DNNs). The presented methods…
This paper presents a distributed Koopman operator learning framework for modeling unknown nonlinear dynamics using sequential observations from multiple agents. Each agent estimates a local Koopman approximation based on lifted data and…
In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the P\'eclet number. In this situation, computational instabilities occur, both for steady and…
Koopman Operator Theory has opened the doors to data-driven learning of globally linear representations of complex nonlinear systems. However, current methodologies for Koopman Operator discovery struggle with uncertainty quantification and…
Petrov-Galerkin formulations with optimal test functions allow for the stabilization of finite element simulations. In particular, given a discrete trial space, the optimal test space induces a numerical scheme delivering the best…
We propose a simple and efficient algorithm for learning sparse invariant representations from unlabeled data with fast inference. When trained on short movies sequences, the learned features are selective to a range of orientations and…
Neural networks transform data through learned representations whose geometry affects separation, contraction, and generalization. Recent work studies this geometry using discrete curvature on neighborhood graphs, suggesting Ricci-flow-like…
Over the last few years, several works have proposed deep learning architectures to learn dynamical systems from observation data with no or little knowledge of the underlying physics. A line of work relies on learning representations where…
This work introduces a kernel-independent, multilevel, adaptive algorithm for efficiently evaluating a discrete convolution kernel with a given source distribution. The method is based on linear algebraic tools such as low rank…
This paper proposes an adaptive hyper-reduction method to reduce the computational cost associated with the simulation of parametric particle-based kinetic plasma models, specifically focusing on the Vlasov-Poisson equation. Conventional…
We present a higher-order boundary condition for atomistic simulations of dislocations that address the slow convergence of standard supercell methods. The method is based on a multipole expansion of the equilibrium displacement, combining…
This letter presents a non-parametric modeling approach for forecasting stochastic dynamical systems on low-dimensional manifolds. The key idea is to represent the discrete shift maps on a smooth basis which can be obtained by the diffusion…
This paper establishes optimal convergence rates for estimation of structured covariance operators of Gaussian processes. We study banded operators with kernels that decay rapidly off-the-diagonal and $L^q$-sparse operators with an…