Eike H. Müller

Eike Hermann Müller

The solution of time dependent differential equations with neural networks has attracted a lot of attention recently. The central idea is to learn the laws that govern the evolution of the solution from data, which might be polluted with random noise. However, in contrast to other machine learning applications, usually a lot is known about the system at hand. For example, for many dynamical systems physical quantities such as energy or (angular) momentum are exactly conserved. Hence, the neural network has to learn these conservation laws from data and they will only be satisfied approximately due to finite training time and random noise. In this paper we present an alternative approach which uses Noether's Theorem to inherently incorporate conservation laws into the architecture of the neural network. We demonstrate that this leads to better predictions for three model systems: the motion of a non-relativistic particle in a three-dimensional Newtonian gravitational potential, the motion of a massive relativistic particle in the Schwarzschild metric and a system of two interacting particles in four dimensions.

S. Aseeri, O. Batrašev, M. Icardi et al.

The cubic Klein-Gordon equation is a simple but non-trivial partial differential equation whose numerical solution has the main building blocks required for the solution of many other partial differential equations. In this study, the library 2DECOMP&FFT is used in a Fourier spectral scheme to solve the Klein-Gordon equation and strong scaling of the code is examined on thirteen different machines for a problem size of 512^3. The results are useful in assessing likely performance of other parallel fast Fourier transform based programs for solving partial differential equations. The problem is chosen to be large enough to solve on a workstation, yet also of interest to solve quickly on a supercomputer, in particular for parametric studies. Unlike other high performance computing benchmarks, for this problem size, the time to solution will not be improved by simply building a bigger supercomputer.

L. M. Kreusser, H. E. Lockyer, E. H. Müller et al.

Splitting methods are widely used for solving initial value problems (IVPs) due to their ability to simplify complicated evolutions into more manageable subproblems which can be solved efficiently and accurately. Traditionally, these methods are derived using analytic and algebraic techniques from numerical analysis, including truncated Taylor series and their Lie algebraic analogue, the Baker--Campbell--Hausdorff formula. These tools enable the development of high-order numerical methods that provide exceptional accuracy for small timesteps. Moreover, these methods often (nearly) conserve important physical invariants, such as mass, unitarity, and energy. However, in many practical applications the computational resources are limited. Thus, it is crucial to identify methods that achieve the best accuracy within a fixed computational budget, which might require taking relatively large timesteps. In this regime, high-order methods derived with traditional methods often exhibit large errors since they are only designed to be asymptotically optimal. Machine Learning techniques offer a potential solution since they can be trained to efficiently solve a given IVP with less computational resources. However, they are often purely data-driven, come with limited convergence guarantees in the small-timestep regime and do not necessarily conserve physical invariants. In this work, we propose a framework for finding machine learned splitting methods that are computationally efficient for large timesteps and have provable convergence and conservation guarantees in the small-timestep limit. We demonstrate numerically that the learned methods, which by construction converge quadratically in the timestep size, can be significantly more efficient than established methods for the Schrödinger equation if the computational budget is limited.

William R. Saunders, James Grant, Eike H. Müller

Molecular Dynamics (MD) codes predict the fundamental properties of matter by following the trajectories of a collection of interacting model particles. To exploit diverse modern manycore hardware, efficient codes must use all available parallelism. At the same time they need to be portable and easily extendible by the domain specialist (physicist/chemist) without detailed knowledge of this hardware. To address this challenge, we recently described a new Domain Specific Language (DSL) for the development of performance portable MD codes based on a "Separation of Concerns": a Python framework automatically generates efficient parallel code for a range of target architectures. Electrostatic interactions between charged particles are important in many physical systems and often dominate the runtime. Here we discuss the inclusion of long-range interaction algorithms in our code generation framework. These algorithms require global communications and careful consideration has to be given to any impact on parallel scalability. We implemented an Ewald summation algorithm for electrostatic forces, present scaling comparisons for different system sizes and compare to the performance of existing codes. We also report on further performance optimisations delivered with OpenMP shared memory parallelism.

William R. Saunders, James Grant, Eike H. Müller

Developers of Molecular Dynamics (MD) codes face significant challenges when adapting existing simulation packages to new hardware. In a continuously diversifying hardware landscape it becomes increasingly difficult for scientists to be experts both in their own domain (physics/chemistry/biology) and specialists in the low level parallelisation and optimisation of their codes. To address this challenge, we describe a "Separation of Concerns" approach for the development of parallel and optimised MD codes: the science specialist writes code at a high abstraction level in a domain specific language (DSL), which is then translated into efficient computer code by a scientific programmer. In a related context, an abstraction for the solution of partial differential equations with grid based methods has recently been implemented in the (Py)OP2 library. Inspired by this approach, we develop a Python code generation system for molecular dynamics simulations on different parallel architectures, including massively parallel distributed memory systems and GPUs. We demonstrate the efficiency of the auto-generated code by studying its performance and scalability on different hardware and compare it to other state-of-the-art simulation packages. With growing data volumes the extraction of physically meaningful information from the simulation becomes increasingly challenging and requires equally efficient implementations. A particular advantage of our approach is the easy expression of such analysis algorithms. We consider two popular methods for deducing the crystalline structure of a material from the local environment of each atom, show how they can be expressed in our abstraction and implement them in the code generation framework.