Neural Wave Functions for the Electronic Schrödinger Equation: A Mathematical Case Study in Scientific Computing.
by
Seminar Room 1.008
Endenicher Allee 60
Deep learning has attracted considerable attention in scientific computing, from neural-network ansätze for partial differential equations to data-driven surrogate models for complex first-principles simulations. Its impact, however, has been uneven: in many standard settings, classical numerical methods remain difficult to outperform. I will begin with a brief broader perspective on this phenomenon, including complexity-theoretic upper and lower bounds that clarify both the limitations of deep-learning-based methods and the special structures under which they can succeed.
The electronic Schrödinger equation provides a particularly compelling example of such a success. In computational quantum chemistry, deep-learning variational Monte Carlo (VMC) has led to striking empirical progress through highly expressive neural-network wave functions. At the same time, this success raises delicate mathematical questions. I will discuss recent results showing that the nodal geometry of the wave function governs the integrability of the local energy and of VMC gradient estimators, leading naturally to heavy-tailed stochastic optimization problems. Motivated by this analysis, I will present a clipped VMC optimization algorithm and prove its convergence under precisely the weak-moment assumptions identified by the nodal theory. The talk will conclude with open questions at the interface of approximation theory, probability, optimization, and computational quantum chemistry.
Alexander Effland, Illia Karabash, Anton Bovier and Christian Brennecke