Researchers from Brown University have built DeepONet, a novel neural network-based model that can efficiently learn both linear and nonlinear operators. This novel model was inspired by earlier studies led by researchers at Fudan University.
A continuous function does not have any abrupt changes in value. More precisely, small changes in continuous function’s output can be assured by restricting to sufficiently small changes in its input. Many studies show that artificial neural networks (ANN) are highly efficient approximators of continuous functions. However, not many studies have yet focused on their ability to approximate nonlinear operators.
Inspired by the papers published by Chen and Chen at Fudan University, which discusses the functional approximation using a single layer of neurons, the researchers decided to explore the possibility of building a neural network that could approximate both linear and nonlinear operators.
Unlike conventional neural networks, which approximate functions, DeepONet approximates both linear and nonlinear operators. This computational model consists of two deep neural networks:
- One deep neural network encodes the discrete input function space.
- The other network encodes the domain of the output functions.
While standard neural networks take data points as inputs and provide data points as outputs, DeepONet takes functions (infinite-dimensional objects) as inputs and maps them to other output space functions. DeepONet’s networks can represent mathematical operators as well as differential equations in continuous output space. Then, DeepONet can complete operations and make faster predictions upon learning the provided operators.
Preliminary evaluation shows that DeepONet can even make predictions related to very complex systems instantly. Additionally, they examined different formulations of DeepONet’s input function space and evaluated these formulations’ impact on the generalization error for 16 other applications. The results proved that the model could implicitly acquire various linear and nonlinear operators.
George Em Karniadakis, one of the researchers, states that with the ability to make predictions in real-time, DeepONet can be extremely useful for autonomous vehicles. It can also be used as a building block to simulate digital twins, systems of systems, and even complex social dynamical systems.
In the future, its possible applications include the development of robots that can solve calculus problems and differential equations and even more responsive and sophisticated autonomous vehicles.
Karniadakis will collaborate with labs from the Department of Energy and other industries to carry out its design applications such as modeling ice-melting in Antarctica, climate modeling, etc.