Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Fast parallel algorithms for short-range molecular dynamics

저작근 근전도에 관한 정상교합자와 ii급 부정교합자의 비교 연구

Despite great progress in simulating multiphysics problems using the numerical discretization of partial differential equations (PDEs), one still cannot seamlessly incorporate noisy data into existing algorithms, mesh generation remains complex, and high-dimensional problems governed by parameterized PDEs cannot be tackled. Moreover, solving inverse problems with hidden physics is often prohibitively expensive and requires different formulations and elaborate computer codes. Machine learning has emerged as a promising alternative, but training deep neural networks requires big data, not always available for scientific problems. Instead, such networks can be trained from additional information obtained by enforcing the physical laws (for example, at random points in the continuous space-time domain). Such physics-informed learning integrates (noisy) data and mathematical models, and implements them through neural networks or other kernel-based regression networks. Moreover, it may be possible to design specialized network architectures that automatically satisfy some of the physical invariants for better accuracy, faster training and improved generalization. Here, we review some of the prevailing trends in embedding physics into machine learning, present some of the current capabilities and limitations and discuss diverse applications of physics-informed learning both for forward and inverse problems, including discovering hidden physics and tackling high-dimensional problems. The rapidly developing field of physics-informed learning integrates data and mathematical models seamlessly, enabling accurate inference of realistic and high-dimensional multiphysics problems. This Review discusses the methodology and provides diverse examples and an outlook for further developments.

Physics-informed machine learning

The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. The Fourier neural operator is the first ML-based method to successfully model turbulent flows with zero-shot super-resolution. It is up to three orders of magnitude faster compared to traditional PDE solvers. Additionally, it achieves superior accuracy compared to previous learning-based solvers under fixed resolution.

Fourier Neural Operator for Parametric Partial Differential Equations

It is widely known that neural networks (NNs) are universal approximators of continuous functions. However, a less known but powerful result is that a NN with a single hidden layer can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is suggestive of the structure and potential of deep neural networks (DNNs) in learning continuous operators or complex systems from streams of scattered data. Here, we thus extend this theorem to DNNs. We design a new network with small generalization error, the deep operator network (DeepONet), which consists of a DNN for encoding the discrete input function space (branch net) and another DNN for encoding the domain of the output functions (trunk net). We demonstrate that DeepONet can learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations. We study different formulations of the input function space and its effect on the generalization error for 16 different diverse applications. Neural networks are known as universal approximators of continuous functions, but they can also approximate any mathematical operator (mapping a function to another function), which is an important capability for complex systems such as robotics control. A new deep neural network called DeepONet can lean various mathematical operators with small generalization error.

Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators

Deep neural operators can learn operators mapping between inﬁnite-dimensional function spaces via deep neural networks and have become an emerging paradigm of scientiﬁc machine learning. However, training neural operators usually requires a large amount of high-ﬁdelity data, which is often diﬃcult to obtain in real engineering problems. Here, we address this challenge by using multiﬁdelity learning, i.e., learning from multiﬁdelity datasets. We develop a multiﬁdelity neural operator based on a deep operator network (DeepONet). A multiﬁdelity DeepONet includes two standard DeepONets coupled by residual learning and input augmentation. Multiﬁdelity DeepONet signiﬁcantly reduces the required amount of high-ﬁdelity data and achieves one order of magnitude smaller error when using the same amount of high-ﬁdelity data. We apply a multiﬁdelity DeepONet to learn the phonon Boltzmann transport equation (BTE), a framework to compute nanoscale heat transport. By combining a trained multiﬁdelity DeepONet with genetic algorithm or topology optimization, we demonstrate a fast solver for the inverse design of BTE problems.

Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport

In this work, the previously developed coarse-grained (CG) particle models for biomembranes and red blood cells (RBCs) are reviewed, and the advantages of the CG particle methods over the continuum and atomistic simulations for modeling biological phenomena are discussed. CG particle models can largely increase the length scale and time scale of atomistic simulations by eliminating the fast degrees of freedom while preserving the mesoscopic structures and properties of the simulated system. Moreover, CG particle models can be used to capture the microstructural alternations in diseased RBCs and simulate the topological changes of biomembranes and RBCs, which are the major challenges to the typical continuum representations of membranes and RBCs. The power and versatility of CG particle methods are demonstrated through simulating the dynamical processes involving significant topological changes, e.g., lipid self-assembly vesicle fusion and membrane budding.

Modeling biomembranes and red blood cells by coarse-grained particle methods

Mesoscopic Adaptive Resolution Scheme toward Understanding of Interactions between Sickle Cell Fibers

Inverse design arises in a variety of areas in engineering such as acoustic, mechanics, thermal/electronic transport, electromagnetism, and optics. Topology optimization is a major form of inverse design, where we optimize a designed geometry to achieve targeted properties and the geometry is parameterized by a density function. This optimization is challenging, because it has a very high dimensionality and is usually constrained by partial differential equations (PDEs) and additional inequalities. Here, we propose a new deep learning method -- physics-informed neural networks with hard constraints (hPINNs) -- for solving topology optimization. hPINN leverages the recent development of PINNs for solving PDEs, and thus does not rely on any numerical PDE solver. However, all the constraints in PINNs are soft constraints, and hence we impose hard constraints by using the penalty method and the augmented Lagrangian method. We demonstrate the effectiveness of hPINN for a holography problem in optics and a fluid problem of Stokes flow. We achieve the same objective as conventional PDE-constrained optimization methods based on adjoint methods and numerical PDE solvers, but find that the design obtained from hPINN is often simpler and smoother for problems whose solution is not unique. Moreover, the implementation of inverse design with hPINN can be easier than that of conventional methods.

Physics-informed neural networks with hard constraints for inverse design

Public good exploitation has been studied extensively from an evolutionary lens, but little is known about the occurrence and impact of public good exploiters in natural communities. Here, we develop a reverse ecology approach to systematically identify bacteria that can exploit public goods produced during the degradation of polysaccharides. Focusing on chitin - the second most abundant biopolymer on the planet, we show that public good exploiters hinder the growth of degraders and invade marine microbial communities during early stages of colonization. Unlike cheaters in social evolution, exploiters and polysaccharide degraders (cooperators) come together by a process of community assembly, belong to distant lineages and can stably coexist. Thus, our approach opens novel avenues to interpret the wealth of genomic data through an ecological lens.

/pdf/public-good-exploitation-in-natural-bacterioplankton-57hdpjzm9v.pdf

Lu Lu

Papers

Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport

Modeling biomembranes and red blood cells by coarse-grained particle methods

Mesoscopic Adaptive Resolution Scheme toward Understanding of Interactions between Sickle Cell Fibers

Physics-informed neural networks with hard constraints for inverse design

Public good exploitation in natural bacterioplankton communities