Uncertainty Quantification for Deep Learning in Materials Design


Eric Hall

School of Mathematics, University of Dundee, United Kingdom

 

Abstract: Modern science and engineering utilize physics-based models to inform decisions and guide design. In these settings, systems of interest are typically complex, exhibiting multiscale/-physics interactions and involving correlated/dependent and non-Gaussian variables. This talk presents two uncertainty quantification technologies for deep learning for physics-based models of complex systems arising in materials design for energy storage. The first is Graph-Informed Neural Networks (GINNs), a strategy for incorporating domain knowledge into machine-learned surrogate models. This framework embeds expert knowledge, available data, and design constraints into a physics-based representation using probabilistic graphical models that provide a context for interpreting the surrogate's predictions, thereby enhancing defensibility. The second is Global Sensitivity Analysis (GSA) based on information theoretic dissimilarity measures. Our information-theoretic GSA provides a model-agnostic uncertainty quantification method for interrogating surrogates compatible with a wide range of black-box models. Effect rankings based on information-theoretic GSA assist in explaining surrogate predictions, thereby enabling deep-learning surrogates to close design loops for rapid simulation-based prototyping. 




First principles electronic friction theory to simulate quantum dynamics at surfaces


Reinhard J. Maurer
 Department of Chemistry and Department of Physics, University of Warwick, United Kingdom


Electronic friction theory is a stochastic mixed quantum-classical simulation method that enables the description of molecular dynamics under the influence of dissipative forces that arise from low-lying electronic excitations. Nonadiabatic effects that arise from the concerted motion of electrons and atoms at comparable energy and time scales are omnipresent in chemical dynamics at metal surfaces. Excited (hot) electrons can measurably affect molecule-metal reactions by contributing to state-dependent reaction probabilities. In this talk, I will present electronic friction theory and how it arises from approximations to time-dependent quantum mechanics. I will describe the mathematical and physical properties of electronic friction and how it affects physically measurable properties. I will further present our recent efforts to combine a tensorial representation of electronic friction based on Functional Theory with high-dimensional machine-learning-based representations of energy and friction landscapes to perform molecular dynamics with electronic friction simulations. Finally, I will conclude with detailed analysis of the successes and limitations of this approach and will highlight the exciting mathematical challenges that lie ahead.



Learnable Equivariant Respresentations of Stochastic Heat Bath Models via the Atomic Cluster Expansion (ACE)


Matthias Sachs

School of Mathematics, University of Birmingham, United Kingdom

Rigorously derived dynamics of coarse-grained particle systems via the Mori-Zwanzig projection formalism take the form of a (generalized) Langevin equation with, in general, configuration-dependent friction and diffusion tensors. In this talk, I will introduce a class of equivariant representations of tensor-valued functions based on the Atomic Cluster Expansion (ACE) framework that allows for efficient learning of such configuration-dependent friction and diffusion tensors from data. Besides satisfying the correct equivariance properties with respect to the Euclidean group E(3), the resulting heat bath models satisfy a fluctuation-dissipation relation. Moreover, such models can be extended to include additional symmetries, such as momentum conservation, to preserve the hydrodynamic properties of the particle system.  






Accurate and Efficient Splitting Methods for Dissipative Particle Dynamics  


Xiaocheng Shang 

School of Mathematics, University of Birmingham


Abstract: We study numerical methods for dissipative particle dynamics (DPD), which is a system of stochastic differential equations and a popular stochastic momentum-conserving thermostat for simulating complex hydrodynamic behavior at mesoscales. We propose a new splitting method that is able to substantially improve the accuracy and efficiency of DPD simulations in a wide range of friction coefficients, particularly in the extremely large friction limit that corresponds to a fluid-like Schmidt number, a key issue in DPD. Various numerical experiments on both equilibrium and transport properties are performed to demonstrate the superiority of the newly proposed method over popular alternative schemes in the literature.



Quantum optimal control


Pranav Singh

School of Mathematics, University of Bath


The various ingredients required for optimal control of physical systems are: (i) efficient numerical solvers that respect properties of the system, (ii) procedures for computation of derivatives, and (iii) optimal control routines that are fast and accurate. The physical systems of interest in this talk are quantum systems, which feature certain challenges unique to these systems. In this talk I will present some recent developments in (i) quantum circuits for approximating quantum dynamics, (ii) Lie algebraic techniques for computing analytic gradients and Hessians of numerical integrators, (iii) an adaptive optimal control procedure called QOALA, and discuss some surprising regularisation properties of certain well known optimisation techniques.


Numerics with Coordinate Transforms for Efficient Brownian Dynamics Simulations


Dominic Phillips

School of Mathematics, University of Birmingham



Many stochastic processes in the physical and biological sciences can be modelled using Brownian dynamics with multiplicative noise. However, numerical integrators for these processes can lose accuracy or even fail to converge when the diffusion term is configuration-dependent. One remedy is to construct a transform to a constant-diffusion process and sample the transformed process instead. In this work, we explain how coordinate-based and time-rescaling-based transforms can be used either individually or in combination to map a general class of variable-diffusion Brownian motion processes into constant-diffusion ones. The transforms are invertible, thus allowing recovery of the original dynamics. We motivate our methodology using examples in one dimension before then considering multivariate diffusion processes. We illustrate the benefits of the transforms through numerical simulations, demonstrating how the right combination of integrator and transform can improve computational efficiency and the order of convergence to the invariant distribution. Notably, the transforms that we derive are applicable to a class of multibody, anisotropic Stokes- Einstein diffusion that has applications in biophysical modelling.