CMX Student/Postdoc Seminar
Approximating Operators with Deep Learning
Efficiently approximating the solution operators arising from parametric PDE systems
is a challenging task with numerous applications in science and engineering. I will present
two recently proposed approaches for this task in a fully data-driven (non-intrusive)
setting. Both follow the philosophy of first conceptualizing an algorithm on the space of
functions then discretizing only when required for computation. This affords rates of approximation
that are independent of the underlying finite-dimensional space used to discretize the data.
The first approach combines ideas from deep learning and projection-based model reduction,
constructing a neural network which links the latent spaces of the input-output snapshots.
The approximation is shown to converge in the limit of infinite data and reduced dimension.
The second approach generalizes standard neural networks defined on finite-dimensional Euclidean
spaces to infinite-dimensional function spaces by replacing the parameter matrix
with a kernel integral operator. A universal approximation result is proved for this
architecture. Numerically, I will demonstrate the efficacy and robustness to discretization
of both approaches on classes of parametric elliptic, parabolic, and hyperbolic PDEs
with applications in underground reservoir modeling, the turbulent flow of fluids,
and the deformation of plastic materials.