While many problems in machine learning focus on learning mappings between finite-dimensional spaces, scientific applications require approximating mappings between function spaces, i.e., operators. We study the problem of learning collections of operators and provide both theoretical and empirical advances. We distinguish between two regimes: (i) multiple operator learning, where a single network represents a continuum of operators parameterized by a parametric function, and (ii) learning several distinct single operators, where each operator is learned independently. For the multiple operator case, we introduce two new architectures, MNO and MONet, and establish universal approximation results in three settings: continuous, integrable, or Lipschitz operators. For the latter, we further derive explicit scaling laws that quantify how the network size must grow to achieve a target approximation accuracy. For learning several single operators, we develop a framework for balancing architectural complexity across subnetworks and show how approximation order determines computational efficiency. Empirical experiments on parametric PDE benchmarks confirm the strong expressive power and efficiency of the proposed architectures. Overall, this work establishes a unified theoretical and practical foundation for scalable neural operator learning across multiple operators.
This paper investigates the role of viscosity in the error upper bounds of a consistent splitting scheme for the Navier-Stokes equations proposed by Huang and Shen [1]. In their original analysis the viscosity is fixed to unity. By following and extending their proof methodology while keeping the viscosity v symbolic, we obtain an H-1 velocity error bound that contains negative powers of v, indicating that the scheme is not robust as v -> 0. To establish this bound we refine a theorem in [2] on the constant in the Stokes pressure estimate, which is crucial to the error analysis. A targeted numerical experiment based on a perturbation of the Kovasznay flow corroborates this analytical prediction: the scheme of [1] blows up at high Reynolds number, and a comparison with a fully implicit Newton solver and with the time-dependent Stokes counterpart of the same scheme localizes the failure to the explicit treatment of the convection term.
We present a data-driven approach for constructing generalized collisional kinetic models for inhomogeneous plasmas in one-dimensional physical space and three-dimensional velocity space (1D-3V). The collision operator is directly learned from micro-scale molecular dynamics (MD) and accurately accounts for the unresolved particle interactions over a broad range of plasma conditions. Unlike the standard Landau operator, the present operator takes an anisotropic, non-stationary form that captures the heterogeneous collisional energy transfer arising from the many-body interactions, which is crucial for plasma kinetics beyond the weakly coupled regime. Efficient numerical evaluation is achieved through a low-rank tensor representation with (9(N log N) computational complexity. The constructed collisional kinetic equation strictly preserves conservation laws and physical constraints at continuum level and therefore, enables us to develop an explicit second-order, energy-conserving scheme that ensures fully discrete conservation of mass and total energy. Numerical results demonstrate that the present model accurately predicts both transport coefficients and several 1D-3V kinetic processes compared with MD simulations across a broad range of densities and temperatures in spatially inhomogeneous settings. This work provides a systematic pathway for bridging micro-scale MD and inhomogeneous plasma kinetic descriptions where empirical models show limitations.
In seismic-prone regions, assessment of the risk of regional bridge networks is essential for quantifying resilience metrics. Ideally this assessment needs to rely on high-fidelity numerical simulation models and on a comprehensive quantification of the seismic hazard. Unfortunately, the computational burden associated with such a faithful assessment of risk across the bridge portfolio is prohibitive for larger networks. This study investigates the use of a Gaussian Process (GP)-based stochastic emulation framework to alleviate this burden. A single surrogate model is established across all bridges of the same typology in the network. The surrogate model development accounts for diverse bridge configurations, as well as aleatoric ground motion variability across different intensity levels. The implementation also addresses aleatoric correlations across different engineering demand parameters for each bridge. The framework is implemented for risk assessment of the Chilean bridge network, specifically for four different bridge typologies, representing the predominant configurations within the national bridge inventory. It also examines in detail the computational benefits of establishing a single surrogate model across each typology, by comparing its accuracy to an application that considers the development of separate surrogate models per bridge type. The case study validates the accuracy of the stochastic emulation framework and illustrates the computational benefits for regional risk assessment. The latter is accomplished by comparing the results to different approximate formulations with respect to the assumptions for modeling the bridges within the network or the fidelity for considering and propagating the different types of uncertainties impacting the risk estimation.
This study introduces a new unstructured computational mesh for hydrodynamic simulations of the Oslofjord. The mesh was created with global bathymetry and shoreline data, using OceanMesh2D. It contains 70,410 nodes, with a resolution at the coastline of 50 m. We use the new mesh to create an advanced circulation model of the fjord. The model is run for four time periods with different characteristics, and validated against the current state of the art and elevation gauges in the fjord.Initial analyses show that the model effectively propagates tides through the narrow channels of the fjord, while requiring minimal computation time. Three different combinations of tidal constituents are used to force the model, and we analyze the cost and benefits of using additional constituents. In order to assess the model in a realistic scenario, the water surface elevation output from a coarse global model is used to force the model for an extreme weather event. The results demonstrate the mesh’s ability to reproduce high-resolution total water levels in the fjord, with a maximum root mean square error of just over 10 cm.
