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Enhanced sampling methods such as metadynamics and umbrella sampling have become essential tools for exploring the configuration space of molecules and materials. At the same time, they have long faced a number of issues such as the inefficiency when dealing with a large number of collective variables (CVs) or systems with high free energy barriers. In this work, we show that with \redc{the clustering and adaptive tuning techniques}, the reinforced dynamics (RiD) scheme can be used to efficiently explore the configuration space and free energy landscapes with a large number of CVs or systems with high free energy barriers. We illustrate this by studying various representative and challenging examples. Firstly we demonstrate the efficiency of adaptive RiD compared with other methods, and construct the 9-dimensional free energy landscape of peptoid trimer which has energy barriers of more than 8 kcal/mol. We then study the folding of the protein chignolin using 18 CVs. In this case, both the folding and unfolding rates are observed to be equal to 4.30~$\mu s^{-1}$. Finally, we propose a protein structure refinement protocol based on RiD. This protocol allows us to efficiently employ more than 100 CVs for exploring the landscape of protein structures and it gives rise to an overall improvement of 14.6 units over the initial Global Distance Test-High Accuracy (GDT-HA) score.

In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality for many different applications and have been proven to be so in the case of some nonlinear Monte Carlo methods for nonlinear parabolic PDEs. In this paper, we review these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the deep backward stochastic differential equations method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying PDEs as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near future.

One of the key issues in the analysis of machine learning models is to identify the appropriate function space and norm for the model. This is the set of functions endowed with a quantity which can control the approximation and estimation errors by a particular machine learning model. In this paper, we address this issue for two representative neural network models: the two-layer networks and the residual neural networks. We define the Barron space and show that it is the right space for two-layer neural network models in the sense that optimal direct and inverse approximation theorems hold for functions in the Barron space. For residual neural network models, we construct the so-called flow-induced function space, and prove direct and inverse approximation theorems for this space. In addition, we show that the Rademacher complexity for bounded sets under these norms has the optimal upper bounds.

The purpose of this article is to review the achievements made in the last few years towards the understanding of the reasons behind the success and subtleties of neural network-based machine learning. In the tradition of good old applied mathematics, we will not only give attention to rigorous mathematical results, but also the insight we have gained from careful numerical experiments as well as the analysis of simplified models. Along the way, we also list the open problems which we believe to be the most important topics for further study. This is not a complete overview over this quickly moving field, but we hope to provide a perspective which may be helpful especially to new researchers in the area., Comment: Review article. Feedback welcome

We develop Banach spaces for ReLU neural networks of finite depth $L$ and infinite width. The spaces contain all finite fully connected $L$-layer networks and their $L^2$-limiting objects under bounds on the natural path-norm. Under this norm, the unit ball in the space for $L$-layer networks has low Rademacher complexity and thus favorable generalization properties. Functions in these spaces can be approximated by multi-layer neural networks with dimension-independent convergence rates. The key to this work is a new way of representing functions in some form of expectations, motivated by multi-layer neural networks. This representation allows us to define a new class of continuous models for machine learning. We show that the gradient flow defined this way is the natural continuous analog of the gradient descent dynamics for the associated multi-layer neural networks. We show that the path-norm increases at most polynomially under this continuous gradient flow dynamics.

To fill the gap between accurate (and expensive) ab initio calculations and efficient atomistic simulations based on empirical interatomic potentials, a new class of descriptions of atomic interactions has emerged and been widely applied; i.e. machine learning potentials (MLPs). One recently developed type of MLP is the deep potential (DP) method. In this review, we provide an introduction to DP methods in computational materials science. The theory underlying the DP method is presented along with a step-by-step introduction to their development and use. We also review materials applications of DPs in a wide range of materials systems. The DP Library provides a platform for the development of DPs and a database of extant DPs. We discuss the accuracy and efficiency of DPs compared with ab initio methods and empirical potentials.

We propose a quantum Monte Carlo approach to solve the ground state many-body Schrodinger equation for the electronic ground state. The method combines optimization from variational Monte Carlo and propagation from auxiliary field quantum Monte Carlo, in a way that significantly alleviates the sign problem. In application to molecular systems, we obtain highly accurate results for configurations dominated by either dynamic or static electronic correlation.

We introduce DeePKS-kit, an open-source software package for developing machine learning based energy and density functional models. DeePKS-kit is interfaced with PyTorch, an open-source machine learning library, and PySCF, an ab initio computational chemistry program that provides simple and customized tools for developing quantum chemistry codes. It supports the DeePHF and DeePKS methods. In addition to explaining the details in the methodology and the software, we also provide an example of developing a chemically accurate model for water clusters.

Machine learning has long been considered as a black box for predicting combustion chemical kinetics due to the extremely large number of parameters and the lack of evaluation standards and reproducibility. The current work aims to understand two basic questions regarding the deep neural network (DNN) method: what data the DNN needs and how general the DNN method can be. Sampling and preprocessing determine the DNN training dataset, further affect DNN prediction ability. The current work proposes using Box-Cox transformation (BCT) to preprocess the combustion data. In addition, this work compares different sampling methods with or without preprocessing, including the Monte Carlo method, manifold sampling, generative neural network method (cycle-GAN), and newly-proposed multi-scale sampling. Our results reveal that the DNN trained by the manifold data can capture the chemical kinetics in limited configurations but cannot remain robust toward perturbation, which is inevitable for the DNN coupled with the flow field. The Monte Carlo and cycle-GAN samplings can cover a wider phase space but fail to capture small-scale intermediate species, producing poor prediction results. A three-hidden-layer DNN, based on the multi-scale method without specific flame simulation data, allows predicting chemical kinetics in various scenarios and being stable during the temporal evolutions. This single DNN is readily implemented with several CFD codes and validated in various combustors, including (1). zero-dimensional autoignition, (2). one-dimensional freely propagating flame, (3). two-dimensional jet flame with triple-flame structure, and (4). three-dimensional turbulent lifted flames. The results demonstrate the satisfying accuracy and generalization ability of the pre-trained DNN. The Fortran and Python versions of DNN and example code are attached in the supplementary for reproducibility.

A new framework is introduced for constructing interpretable and truly reliable reduced models for multiscale problems in situations without scale separation. Hydrodynamic approximation to the kinetic equation is used as an example to illustrate the main steps and issues involved. To this end, a set of generalized moments are constructed first to optimally represent the underlying velocity distribution. The well-known closure problem is then solved with the aim of best capturing the associated dynamics of the kinetic equation. The issue of physical constraints such as Galilean invariance is addressed and an active learning procedure is introduced to help ensure that the dataset used is representative enough. The reduced system takes the form of a conventional moment system and works regardless of the numerical discretization used. Numerical results are presented for the BGK (Bhatnagar--Gross--Krook) model and binary collision of Maxwell molecules. We demonstrate that the reduced model achieves a uniform accuracy in a wide range of Knudsen numbers spanning from the hydrodynamic limit to free molecular flow., Comment: Proceedings of the National Academy of Sciences (2019)

A comprehensive microscopic understanding of ambient liquid water is a major challenge for $ab$ $initio$ simulations as it simultaneously requires an accurate quantum mechanical description of the underlying potential energy surface (PES) as well as extensive sampling of configuration space. Due to the presence of light atoms (e.g., H or D), nuclear quantum fluctuations lead to observable changes in the structural properties of liquid water (e.g., isotope effects), and therefore provide yet another challenge for $ab$ $initio$ approaches. In this work, we demonstrate that the combination of dispersion-inclusive hybrid density functional theory (DFT), the Feynman discretized path-integral (PI) approach, and machine learning (ML) constitutes a versatile $ab$ $initio$ based framework that enables extensive sampling of both thermal and nuclear quantum fluctuations on a quite accurate underlying PES. In particular, we employ the recently developed deep potential molecular dynamics (DPMD) model---a neural-network representation of the $ab$ $initio$ PES---in conjunction with a PI approach based on the generalized Langevin equation (PIGLET) to investigate how isotope effects influence the structural properties of ambient liquid H$_2$O and D$_2$O. Through a detailed analysis of the interference differential cross sections as well as several radial and angular distribution functions, we demonstrate that this approach can furnish a semi-quantitative prediction of these subtle isotope effects., Comment: 19 pages, 5 figures, and 1 table

High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (i) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (ii) a merged formulation of the PDE and the 2BSDE problem, (iii) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (iv) a stochastic gradient descent-type optimization procedure. Numerical results obtained using ${\rm T{\small ENSOR}F{\small LOW}}$ in ${\rm P{\small YTHON}}$ illustrate the efficiency and the accuracy of the method in the cases of a $100$-dimensional Black-Scholes-Barenblatt equation, a $100$-dimensional Hamilton-Jacobi-Bellman equation, and a nonlinear expectation of a $ 100 $-dimensional $ G $-Brownian motion., 56 pages, 12 figures

New estimates for the population risk are established for two-layer neural networks. These estimates are nearly optimal in the sense that the error rates scale in the same way as the Monte Carlo error rates. They are equally effective in the over-parametrized regime when the network size is much larger than the size of the dataset. These new estimates are a priori in nature in the sense that the bounds depend only on some norms of the underlying functions to be fitted, not the parameters in the model, in contrast with most existing results which are a posteriori in nature. Using these a priori estimates, we provide a perspective for understanding why two-layer neural networks perform better than the related kernel methods.

Using the Deep Potential methodology, we construct a model that reproduces accurately the potential energy surface of the SCAN approximation of density functional theory for water, from low temperature and pressure to about 2400 K and 50 GPa, excluding the vapor stability region. The computational efficiency of the model makes it possible to predict its phase diagram using molecular dynamics. Satisfactory overall agreement with experimental results is obtained. The fluid phases, molecular and ionic, and all the stable ice polymorphs, ordered and disordered, are predicted correctly, with the exception of ice III and XV that are stable in experiments, but metastable in the model. The evolution of the atomic dynamics upon heating, as ice VII transforms first into ice VII^{''} and then into an ionic fluid, reveals that molecular dissociation and breaking of the ice rules coexist with strong covalent fluctuations, explaining why only partial ionization was inferred in experiments.

We propose the coarse-grained spectral projection method (CGSP), a deep learning assisted approach for tackling quantum unitary dynamic problems with an emphasis on quench dynamics. We show that CGSP can extract spectral components of many-body quantum states systematically with a sophisticated neural network quantum ansatz. CGSP fully exploits the linear unitary nature of the quantum dynamics and is potentially superior to other quantum Monte Carlo methods for ergodic dynamics. Preliminary numerical results on one-dimensional XXZ models with periodic boundary conditions are carried out to demonstrate the practicality of CGSP.

We propose a general machine learning-based framework for building an accurate and widely-applicable energy functional within the framework of generalized Kohn-Sham density functional theory. To this end, we develop a way of training self-consistent models that are capable of taking large datasets from different systems and different kinds of labels. We demonstrate that the functional that results from this training procedure gives chemically accurate predictions on energy, force, dipole, and electron density for a large class of molecules. It can be continuously improved when more and more data are available.

We introduce a machine-learning-based framework for constructing continuum a non-Newtonian fluid dynamics model directly from a microscale description. Dumbbell polymer solutions are used as examples to demonstrate the essential ideas. To faithfully retain molecular fidelity, we establish a micro-macro correspondence via a set of encoders for the microscale polymer configurations and their macroscale counterparts, a set of nonlinear conformation tensors. The dynamics of these conformation tensors can be derived from the microscale model, and the relevant terms can be parametrized using machine learning. The final model, named the deep non-Newtonian model (DeePN^{2}), takes the form of conventional non-Newtonian fluid dynamics models, with a generalized form of the objective tensor derivative that retains the microscale interpretations. Both the formulation of the dynamic equation and the neural network representation rigorously preserve the rotational invariance, which ensures the admissibility of the constructed model. Numerical results demonstrate the accuracy of DeePN^{2} where models based on empirical closures show limitations.

We characterize the meaning of words with language-independent numerical fingerprints, through a mathematical analysis of recurring patterns in texts. Approximating texts by Markov processes on a long-range time scale, we are able to extract topics, discover synonyms, and sketch semantic fields from a particular document of moderate length, without consulting external knowledge-base or thesaurus. Our Markov semantic model allows us to represent each topical concept by a low-dimensional vector, interpretable as algebraic invariants in succinct statistical operations on the document, targeting local environments of individual words. These language-independent semantic representations enable a robot reader to both understand short texts in a given language (automated question-answering) and match medium-length texts across different languages (automated word translation). Our semantic fingerprints quantify local meaning of words in 14 representative languages across 5 major language families, suggesting a universal and cost-effective mechanism by which human languages are processed at the semantic level. Our protocols and source codes are publicly available on https://github.com/yajun-zhou/linguae-naturalis-principia-mathematica, Comment: Main text (12 pages, 7 figures); Software Manual (ii+262 pages, 16 figures, 12 tables, available as two ancillary files). Revised according to reviewers' comments

Fully anharmonic calculations of the dielectric response of insulators require costly $a\phantom{\rule{0}{0ex}}b$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ molecular dynamics simulations. Here, the authors show that this electronic response property can be described efficiently by a deep neural network that retains the accuracy of $a\phantom{\rule{0}{0ex}}b$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ molecular dynamics. The scheme is demonstrated with calculations of the infrared absorption spectrum of liquid water at standard conditions, and of the evolution of the spectrum of crystalline ice undergoing a pressure-induced structural transformation from molecular ice VII to ionic ice X.

We establish a scale separation of Kolmogorov width type between subspaces of a given Banach space under the condition that a sequence of linear maps converges much faster on one of the subspaces. The general technique is then applied to show that reproducing kernel Hilbert spaces are poor $$L^{2}$$ -approximators for the class of two-layer neural networks in high dimension, and that multi-layer networks with small path norm are poor approximators for certain Lipschitz functions, also in the $$L^{2}$$ -topology.

Neural network-based machine learning is capable of approximating functions in very high dimension with unprecedented efficiency and accuracy. This has opened up many exciting new possibilities, not just in traditional areas of artificial intelligence, but also in scientific computing and computational science. At the same time, machine learning has also acquired the reputation of being a set of "black box" type of tricks, without fundamental principles. This has been a real obstacle for making further progress in machine learning. In this article, we try to address the following two very important questions: (1) How machine learning has already impacted and will further impact computational mathematics, scientific computing and computational science? (2) How computational mathematics, particularly numerical analysis, {can} impact machine learning? We describe some of the most important progress that has been made on these issues. Our hope is to put things into a perspective that will help to integrate machine learning with computational mathematics.

The lack of interpretability and transparency are preventing economists from using advanced tools like neural networks in their empirical research. In this paper, we propose a class of interpretable neural network models that can achieve both high prediction accuracy and interpretability. The model can be written as a simple function of a regularized number of interpretable features, which are outcomes of interpretable functions encoded in the neural network. Researchers can design different forms of interpretable functions based on the nature of their tasks. In particular, we encode a class of interpretable functions named persistent change filters in the neural network to study time series cross-sectional data. We apply the model to predicting individual's monthly employment status using high-dimensional administrative data. We achieve an accuracy of 94.5% in the test set, which is comparable to the best performed conventional machine learning methods. Furthermore, the interpretability of the model allows us to understand the mechanism that underlies the prediction: an individual's employment status is closely related to whether she pays different types of insurances. Our work is a useful step towards overcoming the "black box" problem of neural networks, and provide a new tool for economists to study administrative and proprietary big data.

We present the GPU version of DeePMD-kit, which, upon training a deep neural network model using ab initio data, can drive extremely large-scale molecular dynamics (MD) simulation with ab initio accuracy. Our tests show that the GPU version is 7 times faster than the CPU version with the same power consumption. The code can scale up to the entire Summit supercomputer. For a copper system of 113, 246, 208 atoms, the code can perform one nanosecond MD simulation per day, reaching a peak performance of 86 PFLOPS (43% of the peak). Such unprecedented ability to perform MD simulation with ab initio accuracy opens up the possibility of studying many important issues in materials and molecules, such as heterogeneous catalysis, electrochemical cells, irradiation damage, crack propagation, and biochemical reactions., Comment: 29 pages, 11 figures

We study the natural function space for infinitely wide two-layer neural networks with ReLU activation (Barron space) and establish different representation formulae. In two cases, we describe the space explicitly up to isomorphism. Using a convenient representation, we study the pointwise properties of two-layer networks and show that functions whose singular set is fractal or curved (for example distance functions from smooth submanifolds) cannot be represented by infinitely wide two-layer networks with finite path-norm. We use this structure theorem to show that the only $C^1$-diffeomorphisms which Barron space are affine. Furthermore, we show that every Barron function can be decomposed as the sum of a bounded and a positively one-homogeneous function and that there exist Barron functions which decay rapidly at infinity and are globally Lebesgue-integrable. This result suggests that two-layer neural networks may be able to approximate a greater variety of functions than commonly believed.

We prove that for analytic functions in low dimension, the convergence rate of the deep neural network approximation is exponential.

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black-Scholes equation, the Hamilton-Jacobi-Bellman equation, and the Allen-Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. This opens up new possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their inter-relationships., 13 pages, 6 figures

Recent developments in many-body potential energy representation via deep learning have brought new hopes to addressing the accuracy-versus-efficiency dilemma in molecular simulations. Here we describe DeePMD-kit, a package written in Python/C++ that has been designed to minimize the effort required to build deep learning based representation of potential energy and force field and to perform molecular dynamics. Potential applications of DeePMD-kit span from finite molecules to extended systems and from metallic systems to chemically bonded systems. DeePMD-kit is interfaced with TensorFlow, one of the most popular deep learning frameworks, making the training process highly automatic and efficient. On the other end, DeePMD-kit is interfaced with high-performance classical molecular dynamics and quantum (path-integral) molecular dynamics packages, i.e., LAMMPS and the i-PI, respectively. Thus, upon training, the potential energy and force field models can be used to perform efficient molecular simulations for different purposes. As an example of the many potential applications of the package, we use DeePMD-kit to learn the interatomic potential energy and forces of a water model using data obtained from density functional theory. We demonstrate that the resulted molecular dynamics model reproduces accurately the structural information contained in the original model. Program summary Program Title: DeePMD-kit Program Files doi: http://dx.doi.org/10.17632/hvfh9yvncf.1 Licensing provisions: LGPL Programming language: Python/C++ Nature of problem: Modeling the many-body atomic interactions by deep neural network models. Running molecular dynamics simulations with the models. Solution method: The Deep Potential for Molecular Dynamics (DeePMD) method is implemented based on the deep learning framework TensorFlow. Supports for using a DeePMD model in LAMMPS and i-PI, for classical and quantum (path integral) molecular dynamics are provided. Additional comments including Restrictions and Unusual features: The code defines a data protocol such that the energy, force, and virial calculated by different third-party molecular simulation packages can be easily processed and used as model training data.

We propose a deep learning-based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz Method is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions. The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning. We illustrate the method on several problems including some eigenvalue problems.

We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE. The policy function is then approximated by a neural network, as is done in deep reinforcement learning. Numerical results using TensorFlow illustrate the efficiency and accuracy of the proposed algorithms for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen-Cahn equation, the Hamilton-Jacobi-Bellman equation, and a nonlinear pricing model for financial derivatives., 39 pages, 15 figures

The classic Hegselmann-Krause (HK) model for opinion dynamics consists of a set of agents on the real line, each one instructed to move, at every time step, to the mass center of the agents within a fixed distance R. In this work, we investigate the effects of noise in the continuous-time version of the model as described by its mean-field Fokker-Planck equation. In the presence of a finite number of agents, the system exhibits a phase transition from order to disorder as the noise increases. We introduce an order parameter to track the phase transition and resolve the corresponding phase diagram. The system undergoes a phase transition for small R but none for larger R. Based on the stability analysis of the mean-field equation, we derive the existence of a forbidden zone for the disordered phase to emerge. We also provide a theoretical explanation for the well-known 2R conjecture, which states that, for a random initial distribution in a fixed interval, the final configuration consists of clusters separated by a distance of roughly 2R. Our theoretical analysis confirms previous simulations and predicts properties of the noisy HK model in higher dimension.

We present a continuous formulation of machine learning, as a problem in the calculus of variations and differential-integral equations, in the spirit of classical numerical analysis. We demonstrate that conventional machine learning models and algorithms, such as the random feature model, the two-layer neural network model and the residual neural network model, can all be recovered (in a scaled form) as particular discretizations of different continuous formulations. We also present examples of new models, such as the flow-based random feature model, and new algorithms, such as the smoothed particle method and spectral method, that arise naturally from this continuous formulation. We discuss how the issues of generalization error and implicit regularization can be studied under this framework., published version

A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are updated. General initialization schemes as well as general regimes for the network width and training data size are considered. In the over-parametrized regime, it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the labels. In addition, it is proved that throughout the training process the functions represented by the neural network model are uniformly close to that of a kernel method. For general values of the network width and training data size, sharp estimates of the generalization error is established for target functions in the appropriate reproducing kernel Hilbert space., Published version

An active learning procedure called deep potential generator (DP-GEN) is proposed for the construction of accurate and transferable machine learning-based models of the potential energy surface (PES) for the molecular modeling of materials. This procedure consists of three main components: exploration, generation of accurate reference data, and training. Application to the sample systems of Al, Mg, and Al-Mg alloys demonstrates that DP-GEN can produce uniformly accurate PES models with a minimal number of reference data.