Related papers: Tree-Based Nonlinear Reduced Modeling
Diffusion models have recently emerged as powerful stochastic frameworks for high-dimensional inference and generation. However, existing applications to partial differential equations (PDEs) predominantly rely on physics-informed training…
This paper investigates solution strategies for nonlinear problems in Hilbert spaces, such as nonlinear partial differential equations (PDEs) in Sobolev spaces, when only finite measurements are available. We formulate this as a nonlinear…
We develop a novel computational framework to approximate solution operators of evolution partial differential equations (PDEs). By employing a general nonlinear reduced-order model, such as a deep neural network, to approximate the…
Partial Differential Equations (PDEs) are central to science and engineering. Since solving them is computationally expensive, a lot of effort has been put into approximating their solution operator via both traditional and recently…
Machine learning has been successfully applied to various fields of scientific computing in recent years. In this work, we propose a sparse radial basis function neural network method to solve elliptic partial differential equations (PDEs)…
Physics-based models often involve large systems of parametrized partial differential equations, where design parameters control various properties. However, high-fidelity simulations of such systems on large domains or with high grid…
In this paper, we present new techniques for solving a large variety of partial differential equations. The proposed method reduces the PDEs to first order differential equations known as classical equations such as Bernoulli, Ricatti and…
(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable…
High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the…
This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning…
Projection-based reduced order models are effective at approximating parameter-dependent differential equations that are parametrically separable. When parametric separability is not satisfied, which occurs in both linear and nonlinear…
This work investigates a two-stage method for constructing projection-based reduced-order models (ROMs) of parameterized partial differential equations (PDEs). Based on established tensorial ROM methodology, the proposed approach reduces…
Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature…
Reliable predictions of critical phenomena, such as weather, wildfires and epidemics often rely on models described by Partial Differential Equations (PDEs). However, simulations that capture the full range of spatio-temporal scales…
We present a fully iterative adaptive algorithm for the numerical minimization of strongly convex energy functionals in Hilbert spaces. The proposed approach, which we first present in abstract form, generates a hierarchical sequence of…
We explore how the analysis of the Carleman linearization can be extended to dynamical systems on infinite-dimensional Hilbert spaces with quadratic nonlinearities. We demonstrate the well-posedness and convergence of the truncated Carleman…
Tree structured graphical models are powerful at expressing long range or hierarchical dependency among many variables, and have been widely applied in different areas of computer science and statistics. However, existing methods for…
Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computational domains, etc.…
In this paper, we investigate adaptive nonlinear regression and introduce tree based piecewise linear regression algorithms that are highly efficient and provide significantly improved performance with guaranteed upper bounds in an…
In the theory and practice of inverse problems for partial differential equations (PDEs) much attention is paid to the problem of the identification of coefficients from some additional information. This work deals with the problem of…