Thermostats for sampling and optimization
Benedict Leimkuhler
School of Mathematics, University of Edinburgh
I will discuss the use of thermostat-like temperature controls for wide ranging applications in chemical simulation, statistical computation and machine learning.
Thermostats like Langevin and Nosé-Hoover have their origins in molecular dynamics simulation where they are used to control the thermodynamic state of the system to match a laboratory (or in vivo) setup; as such they are well used tools of materials science. More recently they have been embraced by statisticians who increasingly use them for sampling tasks with respect to general statistical models. They even find uses in machine learning where they help to regulate models containing noisy gradients (as arise when ’subsampling’ is used). In this talk I will discuss two topics in thermostatting: (1) the discretization and parameterization of Langevin dynamics for thermodynamic sampling and (2) the use of Nosé-Hoover-like methods for efficient optimization of general objective functions.
[1] Contraction and convergence rates for discretized kinetic Langevin dynamics, Benedict Leimkuhler, Daniel Paulin, Peter A Whalley, 2023. https://arxiv.org/pdf/2302.10684
[2] Friction-adaptive descent: a family of dynamics-based optimization methods, Katerina Karoni, Benedict Leimkuhler, Gabriel Stoltz, 2023. https://arxiv.org/pdf/2306.06738
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.
Reservoir kernels and Volterra series
Lyudmila Grigoryeva
Faculty of Mathematics and Statistics, University St. Gallen, Switzerland
A universal kernel is constructed whose sections approximate any causal and time-invariant filter in the fading memory category with inputs and outputs in a finite-dimensional Euclidean space. This kernel is built using the reservoir functional associated with a state-space representation of the Volterra series expansion available for any analytic fading memory filter. It is hence called the Volterra reservoir kernel. Even though the state- space representation and the corresponding reservoir feature map are defined on an infinite-dimensional tensor algebra space, the kernel map is characterized by explicit recursions that are readily computable for specific data sets when employed in estimation problems using the representer theorem. We showcase the performance of the Volterra reservoir kernel in a popular data science application in relation to bitcoin price prediction.
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.
Dynamical Properties of Coarse-Grained Stochastic Dynamics
Thomas Hudson
School of Mathematics, University of Warwick
Model-reduction, or coarse-graining, refers to the process of taking a model governing many degrees of freedom and approximating it by a model with a lower number of degrees of freedom. For complex models such as those used in molecular dynamics, the process of coarse-graining enables longer and larger simulations with given computational resources, and so is frequently used by practitioners. Motivated by this application, I will discuss the coarse-graining of linear stochastic dynamical systems, providing a framework within which one may evaluate the accuracy of a coarse-grained model. I will then present recent rigorous mathematical results on the dynamical accuracy of coarse-grained systems in this context obtained with Helen Li.
Hybrid Projection Methods with Recycling for Inverse Problems
Jiahua Jiang
School of Mathematics, University of Birmingham
Iterative hybrid projection methods have proven to be very effective for solving large linear inverse problems due to their inherent regularizing properties as well as the added flexibility to select regularization parameters adaptively. In this work, we develop Golub--Kahan-based hybrid projection methods that can exploit compression and recycling techniques in order to solve a broad class of inverse problems where memory requirements or high computational cost may otherwise be prohibitive. For problems that have many unknown parameters and require many iterations, hybrid projection methods with recycling can be used to compress and recycle the solution basis vectors to reduce the number of solution basis vectors that must be stored, while obtaining a solution accuracy that is comparable to that of standard methods. If reorthogonalization is required, this may also reduce computational cost substantially. In other scenarios, such as streaming data problems or inverse problems with multiple datasets, hybrid projection methods with recycling can be used to efficiently integrate previously computed information for faster and better reconstruction. Additional benefits of the proposed methods are that various subspace selection and compression techniques can be incorporated, standard techniques for automatic regularization parameter selection can be used, and the methods can be applied multiple times in an iterative fashion. Theoretical results show that, under reasonable conditions, regularized solutions for our proposed recycling hybrid method remain close to regularized solutions for standard hybrid methods and reveal important connections among the resulting projection matrices. Numerical examples from image processing show the potential benefits of combining recycling with hybrid projection methods.
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 Edinburgh
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.
An NRDE-based model for solving path-dependent PDEs
Yue Wu
School of Mathematics, Strathclyde University, United Kingdom
The path-dependent partial differential equation (PPDE) was firstly introduced for path-dependent derivatives in the financial market such as Asian, barrier, and lookback options; its semilinear type was later identified as a non-Markovian BSDE. The solution of PPDE contains an infinite-dimensional spatial variable, which makes the solution approximation extremely challenging, if it is not impossible. In this talk, we propose a neural rough differential equation (NRDE) based model to learn (high dimensional) path-dependent parabolic PDEs. This resulting continuous-time model for the PDE solution has the advantage of memory efficiency and coping with variable time-frequency. Several numerical experiments are provided to validate the performance of the proposed model in compassion to strong baselines in the literature. This is joint work with Bowen Fang (University of Warwick, UK) and Hao Ni (UCL, UK).