Related papers: Machine Learning for Auxiliary Sources
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length and timescales. Often, it is computationally intractable to resolve the finest features…
Can neural networks learn to solve partial differential equations (PDEs)? We investigate this question for two (systems of) PDEs, namely, the Poisson equation and the steady Navier--Stokes equations. The contributions of this paper are…
Physics-Informed Neural Networks (PINNs) have become a kind of attractive machine learning method for obtaining solutions of partial differential equations (PDEs). Training PINNs can be seen as a semi-supervised learning task, in which only…
We deal with the numerical solution of linear partial differential equations (PDEs) with focus on the goal-oriented error estimates including algebraic errors arising by an inaccurate solution of the corresponding algebraic systems. The…
Solving partial differential equations (PDEs) is a fundamental problem in science and engineering. While neural PDE solvers can be more efficient than established numerical solvers, they often require large amounts of training data that is…
We explore in detail a method to solve ordinary differential equations using feedforward neural networks. We prove a specific loss function, which does not require knowledge of the exact solution, to be a suitable standard metric to…
Compared to purely data-driven methods, a key feature of physics-informed neural networks (PINNs) - a proven powerful tool for solving partial differential equations (PDEs) - is the embedding of PDE constraints into the loss function. The…
As we know differential equations are very useful for electrical engineers to solve a variety of problems like: voltage across a capacitor, input versus output voltage, etc. Therefore, the goal of this paper is to find the solutions of…
In this paper, we investigate the use of single hidden-layer neural networks as a family of ansatz functions for the resolution of partial differential equations (PDEs). In particular, we train the network via Extreme Learning Machines…
This article investigates the effect of explicitly adding auxiliary algebraic trajectory information to neural networks for dynamical systems. We draw inspiration from the field of differential-algebraic equations and differential equations…
Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and…
With the recent study of deep learning in scientific computation, the Physics-Informed Neural Networks (PINNs) method has drawn widespread attention for solving Partial Differential Equations (PDEs). Compared to traditional methods, PINNs…
Solving partial differential equations (PDEs) using neural networks has become a central focus in scientific machine learning. Training neural networks for singularly perturbed problems is particularly challenging due to certain parameters…
Neural solvers for partial differential equations (PDEs) have great potential to generate fast and accurate physics solutions, yet their practicality is currently limited by their generalizability. PDEs evolve over broad scales and exhibit…
In this work, we present an adaptive adjoint-oriented neural network (adaptive AONN) for solving parametric optimal control problems governed by partial differential equations. The proposed method integrates deep adaptive sampling…
Optimization methods (optimizers) get special attention for the efficient training of neural networks in the field of deep learning. In literature there are many papers that compare neural models trained with the use of different…
Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopted to build the basis of the finite dimensional subspace.…
In this work, we present a parallel scheme for machine learning of partial differential equations. The scheme is based on the decomposition of the training data corresponding to spatial subdomains, where an individual neural network is…
Machine Learning (ML) is increasingly used to construct surrogate models for physical simulations. We take advantage of the ability to generate data using numerical simulations programs to train ML models better and achieve accuracy gain…