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We develop, analyze, and experimentally explore residual-based loss functions for machine learning of parameter-to-solution maps in the context of parameter-dependent families of partial differential equations (PDEs). Our primary concern is…

When using Neural Networks as trial functions to numerically solve PDEs, a key choice to be made is the loss function to be minimised, which should ideally correspond to a norm of the error. In multiple problems, this error norm coincides…

Minimizing PDE-residual losses is a common strategy to promote physical consistency in neural operators. However, standard formulations often lack variational correctness, meaning that small residuals do not guarantee small solution errors…

Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet based technique to solve a coupled system of nonlinear partial differential…

The Deep Fourier Residual (DFR) method is a specific type of variational physics-informed neural networks (VPINNs). It provides a robust neural network-based solution to partial differential equations (PDEs). The DFR strategy is based on…

Residual-based adaptive strategies are widely used in scientific machine learning but remain largely heuristic. We introduce a unifying variational framework that formalizes these methods by integrating convex transformations of the…

Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…

Approximate solutions of partial differential equations (PDEs) obtained by neural networks are highly affected by hyper parameter settings. For instance, the model training strongly depends on loss function design, including the choice of…

We consider the discretization of elliptic boundary-value problems by variational physics-informed neural networks (VPINNs), in which test functions are continuous, piecewise linear functions on a triangulation of the domain. We define an a…

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)…

Stochastic partial differential equations (SPDEs) are ubiquitous in engineering and computational sciences. The stochasticity arises as a consequence of uncertainty in input parameters, constitutive relations, initial/boundary conditions,…

A novel multi-level method for partial differential equations with uncertain parameters is proposed. The principle behind the method is that the error between grid levels in multi-level methods has a spatial structure that is by good…

We present a contribution to the field of system identification of partial differential equations (PDEs), with emphasis on discerning between competing mathematical models of pattern-forming physics. The motivation comes from developmental…

We study the parameter estimation for parabolic, linear, second-order, stochastic partial differential equations (SPDEs) observing a mild solution on a discrete grid in time and space. A high-frequency regime is considered where the mesh of…

Pattern formation is a widely observed phenomenon in diverse fields including materials physics, developmental biology and ecology, among many others. The physics underlying the patterns is specific to the mechanisms, and is encoded by…

Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error,…

In this paper we propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks (NNs). Particularly, we propose a novel loss function that relies…

To solve high-dimensional parameter-dependent partial differential equations (pPDEs), a neural network architecture is presented. It is constructed to map parameters of the model data to corresponding finite element solutions. To improve…

In this paper, we propose a new approach to model reduction of parameterized partial differential equations (PDEs) based on the concept of adaptive reduced bases. The presented approach is particularly suited for large-scale nonlinear…

We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We…