Christoph Ortner

Hybrid Mechanistic & Data-driven Modelling

In mathematical modelling it has always been common practise to construct constitutive laws, e.g., the driving force of a dynamical system

x˙=f(x) \dot{x} = f(x)

by choosing a (well-motivated) functional form f(x)=F(θ,x)f(x) = F(\theta, x) with free parameters θ\theta and fit these parameters to theoretical predictions and/or experimental observations. With the explosion of machine learning it has now become easy to fit general functions ff with even O(100k)O(100k) or more parameters. A common situation is that xRdx \in \mathbb{R}^d with dd large, and where ff may be fitted to a high fidelity model f~\tilde{f} which is very costly to evaluate. In this setting, the problem of fitting ff~f \approx \tilde{f} becomes a classical high-dimensional approximation problem. Coarse-graining may simply mean reducing the computational cost, or more ambitiously one may seek to reduce the dimensionality of the system.

Learning the ab initio potential energy landscape

The context in which these kinds of questions interest me is that we are given an electronic structure model for a potential energy surface,

EQM({rj}j=1J)=F({rj}j=1J;{ϵi}i,{ψi}i) E^{\rm QM}\Big( \{ {\bf r}_j \}_{j = 1}^J \Big) = \mathcal{F}\Big( \{ {\bf r}_{j} \}_{j = 1}^J ; \{ \epsilon_i \}_i, \{ \psi_i \}_i \Big)

where the evaluation of EQME^{\rm QM} involves the solution of a large-scale nonlinear eigenvalue problem for the energy levels ϵi\epsilon_i occupied by the electrons ψi\psi_i with at least O(J3)O(J^3) scaling cost, but depending on the level of theory, it could be even higher up to exponential scaling for the full Schrödinger equation.

There has been tremendous progress in the computational physics and chemistry community in developing machine learning models, usually based on GP regression or ANNs, though even classical polynomials provide excellent models: MTPs or ACE. The overarching strategy (with some variations) is to first write a model PES as

E({r_j}j=1J)=i=1JV({rjri}ji) E\Big( \{ {\bf r}\_j \}_{j = 1}^J \Big) = \sum_{i = 1}^J V\Big( \{ {\bf r}_j - {\bf r}_i \}_{j \neq i} \Big)

where V=V({rij})V = V( \{ {\bf r}_{ij} \} ) is a site energy potential that must be invariant under isometries and permutations of like atoms. There are several challenges that make the modelling of VV a rich mathematical problem, ranging from the analysis of the quantum PES, to the optimal exploitation of symmetries in approximation and regression schemes and finally the fact that the data from which the potentials are fitted is always incomplete which makes this a difficult inverse problem (even in the absence of noise).

Some concrete topics that interest me at the moment: