Related papers: Learning Subgrid-scale Models with Neural Ordinary…
Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…
Learning continuous-time dynamics on complex networks is crucial for understanding, predicting and controlling complex systems in science and engineering. However, this task is very challenging due to the combinatorial complexities in the…
Domain decomposition methods (DDMs) are popular solvers for discretized systems of partial differential equations (PDEs), with one-level and multilevel variants. These solvers rely on several algorithmic and mathematical parameters,…
The advancement of human healthspan and bioengineering relies heavily on predicting the behavior of complex biological systems. While high-throughput multiomics data is becoming increasingly abundant, converting this data into actionable…
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…
Partial differential equations (PDEs) are widely used for modeling various physical phenomena. These equations often depend on certain parameters, necessitating either the identification of optimal parameters or the solution of the…
We develop a novel deep learning technique, termed Deep Orthogonal Decomposition (DOD), for dimensionality reduction and reduced order modeling of parameter dependent partial differential equations. The approach consists in the construction…
This research introduces an extended application of neural networks for solving nonlinear partial differential equations (PDEs). A neural network, combined with a pseudo-arclength continuation, is proposed to construct bifurcation diagrams…
Neural Ordinary Differential Equations (N-ODEs) are a powerful building block for learning systems, which extend residual networks to a continuous-time dynamical system. We propose a Bayesian version of N-ODEs that enables well-calibrated…
We present Neural Spectral Methods, a technique to solve parametric Partial Differential Equations (PDEs), grounded in classical spectral methods. Our method uses orthogonal bases to learn PDE solutions as mappings between spectral…
Identifying parameters in partial differential equations (PDEs) represents a very broad class of applied inverse problems. In recent years, several unsupervised learning approaches using (deep) neural networks have been developed to solve…
In this paper, we address the issue of modeling and estimating changes in the state of the spatio-temporal dynamical systems based on a sequence of observations like video frames. Traditional numerical simulation systems depend largely on…
We propose a new reduced order modeling strategy for tackling parametrized Partial Differential Equations (PDEs) with linear constraints, in particular Darcy flow systems in which the constraint is given by mass conservation. Our approach…
Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In…
We present a new scientific machine learning method that learns from data a computationally inexpensive surrogate model for predicting the evolution of a system governed by a time-dependent nonlinear partial differential equation (PDE), an…
The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin.…
Unlike conventional grid and mesh based methods for solving partial differential equations (PDEs), neural networks have the potential to break the curse of dimensionality, providing approximate solutions to problems where using classical…
We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate…
The recently developed physics-informed machine learning has made great progress for solving nonlinear partial differential equations (PDEs), however, it may fail to provide reasonable approximations to the PDEs with discontinuous…
In this paper, we propose a generic algorithm to train machine learning-based subgrid parametrizations online, i.e., with \textit{a posteriori} loss functions, but for non-differentiable numerical solvers. The proposed approach leverages a…