Related papers: Derivative-Informed Neural Operator: An Efficient …
In multi-body dynamics, the motion of a complicated physical object is described as a coupled ordinary differential equation system with multiple unknown solutions. Engineers need to constantly adjust the object to meet requirements at the…
Time-dependent partial differential equations are ubiquitous in physics-based modeling, but they remain computationally intensive in many-query scenarios, such as real-time forecasting, optimal control, and uncertainty quantification.…
Derivative-free optimization (DFO) consists in finding the best value of an objective function without relying on derivatives. To tackle such problems, one may build approximate derivatives, using for instance finite-difference estimates.…
Regularization plays a pivotal role in integrating prior information into inverse problems. While many deep learning methods have been proposed to solve inverse problems, determining where to apply regularization remains a crucial…
Neural operator models for solving partial differential equations (PDEs) often rely on global mixing mechanisms-such as spectral convolutions or attention-which tend to oversmooth sharp local dynamics and introduce high computational cost.…
Physics-informed neural operators (PINOs) have emerged as powerful tools for learning solution operators of partial differential equations (PDEs). Recent research has demonstrated that incorporating Lie point symmetry information can…
A core challenge in the interpretation of deep neural networks is identifying commonalities between the underlying algorithms implemented by distinct networks trained for the same task. Motivated by this problem, we introduce DYNAMO, an…
Recent advances in the theory of Neural Operators (NOs) have enabled fast and accurate computation of the solutions to complex systems described by partial differential equations (PDEs). Despite their great success, current NO-based…
Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order…
Optimal control problems with nonsmooth objectives and nonlinear partial differential equation (PDE) constraints are challenging, mainly because of the underlying nonsmooth and nonconvex structures and the demanding computational cost for…
A computed approximation of the solution operator to a system of partial differential equations (PDEs) is needed in various areas of science and engineering. Neural operators have been shown to be quite effective at predicting these…
Neural operator learning directly constructs the mapping relationship from the equation parameter space to the solution space, enabling efficient direct inference in practical applications without the need for repeated solution of partial…
This focused review explores a range of neural operator architectures for approximating solutions to parametric partial differential equations (PDEs), emphasizing high-level concepts and practical implementation strategies. The study covers…
Designing universal artificial intelligence (AI) solver for partial differential equations (PDEs) is an open-ended problem and a significant challenge in science and engineering. Currently, data-driven solvers have achieved great success,…
We develop a new and general encode-approximate-reconstruct operator learning model that leverages learned neural representations of bases for input and output function distributions. We introduce the concepts of \textit{numerical operator…
We study the geometry of deep (neural) networks (DNs) with piecewise affine and convex nonlinearities. The layers of such DNs have been shown to be {\em max-affine spline operators} (MASOs) that partition their input space and apply a…
A practical limitation of deep neural networks is their high degree of specialization to a single task and visual domain. Recently, inspired by the successes of transfer learning, several authors have proposed to learn instead universal,…
Acquisition of large datasets for three-dimensional (3D) partial differential equations (PDE) is usually very expensive. Physics-informed neural operator (PINO) eliminates the high costs associated with generation of training datasets, and…
We build a rigorous bridge between deep networks (DNs) and approximation theory via spline functions and operators. Our key result is that a large class of DNs can be written as a composition of max-affine spline operators (MASOs), which…
We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms. A comparative analysis between…