Neural networks have shown great success at approximating functions between spaces X and Y, in the setting where X is a finite dimensional Euclidean space and where Y is either a finite dimensional Euclidean space (regression) or a set of finite cardinality (classification); the neural networks learn the approximator from N data pairs {x_n, y_n}.
In many problems arising in PDEs it is desirable to learn solution operators: maps between spaces of functions X and Y; here X denotes a function space of inputs to the PDE (such as initial conditions, boundary data, coefficients) and Y denotes the function space of PDE solutions. Such a learned map can provide a cheap surrogate model to accelerate computations.
The talk overviews the methodology being developed in this field of operator learning and describes analysis of the associated approximation theory. Applications are described to the learning of homogenized constitutive models in mechanics.