One of the most exciting applications of artificial intelligence (AI) is automated scientific discovery based on previously amassed data, coupled with restrictions provided by known physical principles, including symmetries and conservation laws. Such automated hypothesis creation and verification can assist scientists in studying complex phenomena, where traditional physical intuition may fail. Here we develop a platform based on a generalized Onsager principle to learn macroscopic dynamical descriptions of arbitrary stochastic dissipative systems directly from observations of their microscopic trajectories. Our method simultaneously constructs reduced thermodynamic coordinates and interprets the dynamics on these coordinates. We demonstrate its effectiveness by studying theoretically and validating experimentally the stretching of long polymer chains in an externally applied field. Specifically, we learn three interpretable thermodynamic coordinates and build a dynamical landscape of polymer stretching, including the identification of stable and transition states and the control of the stretching rate. Our general methodology can be used to address a wide range of scientific and technological applications.
Alexander E. Siemenn, Zekun Ren, Qianxiao Li et al.
Needle-in-a-Haystack problems exist across a wide range of applications including rare disease prediction, ecological resource management, fraud detection, and material property optimization. A Needle-in-a-Haystack problem arises when there is an extreme imbalance of optimum conditions relative to the size of the dataset. For example, only $0.82\%$ out of $146$k total materials in the open-access Materials Project database have a negative Poisson's ratio. However, current state-of-the-art optimization algorithms are not designed with the capabilities to find solutions to these challenging multidimensional Needle-in-a-Haystack problems, resulting in slow convergence to a global optimum or pigeonholing into a local minimum. In this paper, we present a Zooming Memory-Based Initialization algorithm, entitled ZoMBI. ZoMBI actively extracts knowledge from the previously best-performing evaluated experiments to iteratively zoom in the sampling search bounds towards the global optimum "needle" and then prunes the memory of low-performing historical experiments to accelerate compute times by reducing the algorithm time complexity from $O(n^3)$ to $O(φ^3)$ for $φ$ forward experiments per activation, which trends to a constant $O(1)$ over several activations. Additionally, ZoMBI implements two custom adaptive acquisition functions to further guide the sampling of new experiments toward the global optimum. We validate the algorithm's optimization performance on three real-world datasets exhibiting Needle-in-a-Haystack and further stress-test the algorithm's performance on an additional 174 analytical datasets. The ZoMBI algorithm demonstrates compute time speed-ups of 400x compared to traditional Bayesian optimization as well as efficiently discovering optima in under 100 experiments that are up to 3x more highly optimized than those discovered by similar methods MiP-EGO, TuRBO, and HEBO.
Danimir T. Doncevic, Alexander Mitsos, Yue Guo et al.
Meta-learning of numerical algorithms for a given task consists of the data-driven identification and adaptation of an algorithmic structure and the associated hyperparameters. To limit the complexity of the meta-learning problem, neural architectures with a certain inductive bias towards favorable algorithmic structures can, and should, be used. We generalize our previously introduced Runge-Kutta neural network to a recursively recurrent neural network (R2N2) superstructure for the design of customized iterative algorithms. In contrast to off-the-shelf deep learning approaches, it features a distinct division into modules for generation of information and for the subsequent assembly of this information towards a solution. Local information in the form of a subspace is generated by subordinate, inner, iterations of recurrent function evaluations starting at the current outer iterate. The update to the next outer iterate is computed as a linear combination of these evaluations, reducing the residual in this space, and constitutes the output of the network. We demonstrate that regular training of the weight parameters inside the proposed superstructure on input/output data of various computational problem classes yields iterations similar to Krylov solvers for linear equation systems, Newton-Krylov solvers for nonlinear equation systems, and Runge-Kutta integrators for ordinary differential equations. Due to its modularity, the superstructure can be readily extended with functionalities needed to represent more general classes of iterative algorithms traditionally based on Taylor series expansions.
Siyu Isaac Parker Tian, Zekun Ren, Selvaraj Venkataraj et al.
Transfer learning increasingly becomes an important tool in handling data scarcity often encountered in machine learning. In the application of high-throughput thickness as a downstream process of the high-throughput optimization of optoelectronic thin films with autonomous workflows, data scarcity occurs especially for new materials. To achieve high-throughput thickness characterization, we propose a machine learning model called thicknessML that predicts thickness from UV-Vis spectrophotometry input and an overarching transfer learning workflow. We demonstrate the transfer learning workflow from generic source domain of generic band-gapped materials to specific target domain of perovskite materials, where the target domain data only come from limited number (18) of refractive indices from literature. The target domain can be easily extended to other material classes with a few literature data. Defining thickness prediction accuracy to be within-10% deviation, thicknessML achieves 92.2% (with a deviation of 3.6%) accuracy with transfer learning compared to 81.8% (with a deviation of 3.6%) 11.7% without (lower mean and larger standard deviation). Experimental validation on six deposited perovskite films also corroborates the efficacy of the proposed workflow by yielding a 10.5% mean absolute percentage error (MAPE).
Despite the wide applications of neural networks, there have been increasing concerns about their vulnerability issue. While numerous attack and defense techniques have been developed, this work investigates the robustness issue from a new angle: can we design a self-healing neural network that can automatically detect and fix the vulnerability issue by itself? A typical self-healing mechanism is the immune system of a human body. This biology-inspired idea has been used in many engineering designs but is rarely investigated in deep learning. This paper considers the post-training self-healing of a neural network, and proposes a closed-loop control formulation to automatically detect and fix the errors caused by various attacks or perturbations. We provide a margin-based analysis to explain how this formulation can improve the robustness of a classifier. To speed up the inference of the proposed self-healing network, we solve the control problem via improving the Pontryagin Maximum Principle-based solver. Lastly, we present an error estimation of the proposed framework for neural networks with nonlinear activation functions. We validate the performance on several network architectures against various perturbations. Since the self-healing method does not need a-priori information about data perturbations/attacks, it can handle a broad class of unforeseen perturbations.
Stephen G. Dale, Nikita Kazeev, Alastair J. A. Price et al.
Artificial intelligence and machine learning are reshaping how we approach scientific discovery, not by replacing established methods but by extending what researchers can probe, predict, and design. In this roadmap we provide a forward-looking view of AI-enabled science across biology, chemistry, climate science, mathematics, materials science, physics, self-driving laboratories and unconventional computing. Several shared themes emerge: the need for diverse and trustworthy data, transferable electronic-structure and interatomic models, AI systems integrated into end-to-end scientific workflows that connect simulations to experiments and generative systems grounded in synthesisability rather than purely idealised phases. Across domains, we highlight how large foundation models, active learning and self-driving laboratories can close loops between prediction and validation while maintaining reproducibility and physical interpretability. Taken together, these perspectives outline where AI-enabled science stands today, identify bottlenecks in data, methods and infrastructure, and chart concrete directions for building AI systems that are not only more powerful but also more transparent and capable of accelerating discovery in complex real-world environments.
In this paper, we investigate the long-term memory learning capabilities of state-space models (SSMs) from the perspective of parameterization. We prove that state-space models without any reparameterization exhibit a memory limitation similar to that of traditional RNNs: the target relationships that can be stably approximated by state-space models must have an exponential decaying memory. Our analysis identifies this "curse of memory" as a result of the recurrent weights converging to a stability boundary, suggesting that a reparameterization technique can be effective. To this end, we introduce a class of reparameterization techniques for SSMs that effectively lift its memory limitations. Besides improving approximation capabilities, we further illustrate that a principled choice of reparameterization scheme can also enhance optimization stability. We validate our findings using synthetic datasets, language models and image classifications.
We investigate the expressive power of deep residual neural networks idealized as continuous dynamical systems through control theory. Specifically, we consider two properties that arise from supervised learning, namely universal interpolation - the ability to match arbitrary input and target training samples - and the closely related notion of universal approximation - the ability to approximate input-target functional relationships via flow maps. Under the assumption of affine invariance of the control family, we give a characterisation of universal interpolation, showing that it holds for essentially any architecture with non-linearity. Furthermore, we elucidate the relationship between universal interpolation and universal approximation in the context of general control systems, showing that the two properties cannot be deduced from each other. At the same time, we identify conditions on the control family and the target function that ensures the equivalence of the two notions.
We study the approximation of functions which are invariant with respect to certain permutations of the input indices using flow maps of dynamical systems. Such invariant functions includes the much studied translation-invariant ones involving image tasks, but also encompasses many permutation-invariant functions that finds emerging applications in science and engineering. We prove sufficient conditions for universal approximation of these functions by a controlled equivariant dynamical system, which can be viewed as a general abstraction of deep residual networks with symmetry constraints. These results not only imply the universal approximation for a variety of commonly employed neural network architectures for symmetric function approximation, but also guide the design of architectures with approximation guarantees for applications involving new symmetry requirements.
We study the approximation of shift-invariant or equivariant functions by deep fully convolutional networks from the dynamical systems perspective. We prove that deep residual fully convolutional networks and their continuous-layer counterpart can achieve universal approximation of these symmetric functions at constant channel width. Moreover, we show that the same can be achieved by non-residual variants with at least 2 channels in each layer and convolutional kernel size of at least 2. In addition, we show that these requirements are necessary, in the sense that networks with fewer channels or smaller kernels fail to be universal approximators.
Despite the effectiveness of deep neural networks in numerous natural language processing applications, recent findings have exposed the vulnerability of these language models when minor perturbations are introduced. While appearing semantically indistinguishable to humans, these perturbations can significantly reduce the performance of well-trained language models, raising concerns about the reliability of deploying them in safe-critical situations. In this work, we construct a computationally efficient self-healing process to correct undesired model behavior during online inference when perturbations are applied to input data. This is formulated as a trajectory optimization problem in which the internal states of the neural network layers are automatically corrected using a PID (Proportional-Integral-Derivative) control mechanism. The P controller targets immediate state adjustments, while the I and D controllers consider past states and future dynamical trends, respectively. We leverage the geometrical properties of the training data to design effective linear PID controllers. This approach reduces the computational cost to that of using just the P controller, instead of the full PID control. Further, we introduce an analytical method for approximating the optimal control solutions, enhancing the real-time inference capabilities of this controlled system. Moreover, we conduct a theoretical error analysis of the analytic solution in a simplified setting. The proposed PID control-based self-healing is a low cost framework that improves the robustness of pre-trained large language models, whether standard or robustly trained, against a wide range of perturbations. A detailed implementation can be found in:https://github.com/zhuotongchen/PID-Control-Based-Self-Healing-to-Improve-the-Robustness-of-Large-Language-Models.
Iterative methods are ubiquitous in large-scale scientific computing applications, and a number of approaches based on meta-learning have been recently proposed to accelerate them. However, a systematic study of these approaches and how they differ from meta-learning is lacking. In this paper, we propose a framework to analyze such learning-based acceleration approaches, where one can immediately identify a departure from classical meta-learning. We show that this departure may lead to arbitrary deterioration of model performance. Based on our analysis, we introduce a novel training method for learning-based acceleration of iterative methods. Furthermore, we theoretically prove that the proposed method improves upon the existing methods, and demonstrate its significant advantage and versatility through various numerical applications.
The performance of state-of-the-art machine learning models often deteriorates when testing on demographics that are under-represented in the training dataset. This problem has predominately been studied in a supervised learning setting where the data distribution is static. However, real-world applications often involve distribution shifts caused by the deployed models. For instance, the performance disparity against monitory users can lead to a high customer churn rate, thus the available data provided by active users are skewed due to the lack of minority users. This feedback effect further exacerbates the disparity among different demographic groups in future steps. To address this issue, we propose asymptotically fair participation as a condition to maintain long-term model performance over all demographic groups. In this work, we aim to address the problem of achieving asymptotically fair participation via optimal control formulation. Moreover, we design a surrogate retention system based on existing literature on evolutionary population dynamics to approximate the dynamics of distribution shifts on active user counts, from which the objective of achieving asymptotically fair participation is formulated as an optimal control problem, and the control variables are considered as the model parameters. We apply an efficient implementation of Pontryagin's maximum principle to estimate the optimal control solution. To evaluate the effectiveness of the proposed method, we design a generic simulation environment that simulates the population dynamics of the feedback effect between user retention and model performance. When we deploy the resulting models to the simulation environment, the optimal control solution accounts for long-term planning and leads to superior performance compared with existing baseline methods.
Scientific computing is an essential tool for scientific discovery and engineering design, and its computational cost is always a main concern in practice. To accelerate scientific computing, it is a promising approach to use machine learning (especially meta-learning) techniques for selecting hyperparameters of traditional numerical methods. There have been numerous proposals to this direction, but many of them require automatic-differentiable numerical methods. However, in reality, many practical applications still depend on well-established but non-automatic-differentiable legacy codes, which prevents practitioners from applying the state-of-the-art research to their own problems. To resolve this problem, we propose a non-intrusive methodology with a novel gradient estimation technique to combine machine learning and legacy numerical codes without any modification. We theoretically and numerically show the advantage of the proposed method over other baselines and present applications of accelerating established non-automatic-differentiable numerical solvers implemented in PETSc, a widely used open-source numerical software library.
We prove an inverse approximation theorem for the approximation of nonlinear sequence-to-sequence relationships using recurrent neural networks (RNNs). This is a so-called Bernstein-type result in approximation theory, which deduces properties of a target function under the assumption that it can be effectively approximated by a hypothesis space. In particular, we show that nonlinear sequence relationships that can be stably approximated by nonlinear RNNs must have an exponential decaying memory structure - a notion that can be made precise. This extends the previously identified curse of memory in linear RNNs into the general nonlinear setting, and quantifies the essential limitations of the RNN architecture for learning sequential relationships with long-term memory. Based on the analysis, we propose a principled reparameterization method to overcome the limitations. Our theoretical results are confirmed by numerical experiments. The code has been released in https://github.com/radarFudan/Curse-of-memory
A State Space Model (SSM) is a foundation model in time series analysis, which has recently been shown as an alternative to transformers in sequence modeling. In this paper, we theoretically study the generalization of SSMs and propose improvements to training algorithms based on the generalization results. Specifically, we give a \textit{data-dependent} generalization bound for SSMs, showing an interplay between the SSM parameters and the temporal dependencies of the training sequences. Leveraging the generalization bound, we (1) set up a scaling rule for model initialization based on the proposed generalization measure, which significantly improves the robustness of the output value scales on SSMs to different temporal patterns in the sequence data; (2) introduce a new regularization method for training SSMs to enhance the generalization performance. Numerical results are conducted to validate our results.
Learning unknown stochastic differential equations (SDEs) from observed data is a significant and challenging task with applications in various fields. Current approaches often use neural networks to represent drift and diffusion functions, and construct likelihood-based loss by approximating the transition density to train these networks. However, these methods often rely on one-step stochastic numerical schemes, necessitating data with sufficiently high time resolution. In this paper, we introduce novel approximations to the transition density of the parameterized SDE: a Gaussian density approximation inspired by the random perturbation theory of dynamical systems, and its extension, the dynamical Gaussian mixture approximation (DynGMA). Benefiting from the robust density approximation, our method exhibits superior accuracy compared to baseline methods in learning the fully unknown drift and diffusion functions and computing the invariant distribution from trajectory data. And it is capable of handling trajectory data with low time resolution and variable, even uncontrollable, time step sizes, such as data generated from Gillespie's stochastic simulations. We then conduct several experiments across various scenarios to verify the advantages and robustness of the proposed method.
Current methods for initializing state space model (SSM) parameters primarily rely on the HiPPO framework \citep{gu2023how}, which is based on online function approximation with the SSM kernel basis. However, the HiPPO framework does not explicitly account for the effects of the temporal structures of input sequences on the optimization of SSMs. In this paper, we take a further step to investigate the roles of SSM initialization schemes by considering the autocorrelation of input sequences. Specifically, we: (1) rigorously characterize the dependency of the SSM timescale on sequence length based on sequence autocorrelation; (2) find that with a proper timescale, allowing a zero real part for the eigenvalues of the SSM state matrix mitigates the curse of memory while still maintaining stability at initialization; (3) show that the imaginary part of the eigenvalues of the SSM state matrix determines the conditioning of SSM optimization problems, and uncover an approximation-estimation tradeoff when training SSMs with a specific class of target functions.
Macroscopic observables of a system are of keen interest in real applications such as the design of novel materials. Current methods rely on microscopic trajectory simulations, where the forces on all microscopic coordinates need to be computed or measured. However, this can be computationally prohibitive for realistic systems. In this paper, we propose a method to learn macroscopic dynamics requiring only force computations on a subset of the microscopic coordinates. Our method relies on a sparsity assumption: the force on each microscopic coordinate relies only on a small number of other coordinates. The main idea of our approach is to map the training procedure on the macroscopic coordinates back to the microscopic coordinates, on which partial force computations can be used as stochastic estimation to update model parameters. We provide a theoretical justification of this under suitable conditions. We demonstrate the accuracy, force computation efficiency, and robustness of our method on learning macroscopic closure models from a variety of microscopic systems, including those modeled by partial differential equations or molecular dynamics simulations.
Mengyi Chen, Pengru Huang, Kostya S. Novoselov et al.
Macroscopic dynamical descriptions of complex physical systems are crucial for understanding and controlling material behavior. With the growing availability of data and compute, machine learning has become a promising alternative to first-principles methods to build accurate macroscopic models from microscopic trajectory simulations. However, for spatially extended systems, direct simulations of sufficiently large microscopic systems that inform macroscopic behavior is prohibitive. In this work, we propose a framework that learns the macroscopic dynamics of large stochastic microscopic systems using only small-system simulations. Our framework employs a partial evolution scheme to generate training data pairs by evolving large-system snapshots within local patches. We subsequently identify the closure variables associated with the macroscopic observables and learn the macroscopic dynamics using a custom loss. Furthermore, we introduce a hierarchical upsampling scheme that enables efficient generation of large-system snapshots from small-system trajectory distributions. We empirically demonstrate the accuracy and robustness of our framework through a variety of stochastic spatially extended systems, including those described by stochastic partial differential equations, idealised lattice spin systems, and a more realistic NbMoTa alloy system.
We consider learning a predictive model to be subsequently used for a given downstream task (described by an algorithm) that requires access to the model evaluation. This task need not be prediction, and this situation is frequently encountered in machine-learning-augmented scientific computing. We show that this setting differs from classical supervised learning, and in general it cannot be solved by minimizing the mean square error of the model predictions as is frequently performed in the literature. Instead, we find that the maximum prediction error on the support of the downstream task algorithm can serve as an effective estimate for the subsequent task performance. With this insight, we formulate a task-specific supervised learning problem based on the given sampling measure, whose solution serves as a reliable surrogate model for the downstream task. Then, we discretize the empirical risk based on training data, and develop an iterative algorithm to solve the task-specific supervised learning problem. Three illustrative numerical examples on trajectory prediction, optimal control and minimum energy path computation demonstrate the effectiveness of the approach.
We present a theoretical analysis of the approximation properties of convolutional architectures when applied to the modeling of temporal sequences. Specifically, we prove an approximation rate estimate (Jackson-type result) and an inverse approximation theorem (Bernstein-type result), which together provide a comprehensive characterization of the types of sequential relationships that can be efficiently captured by a temporal convolutional architecture. The rate estimate improves upon a previous result via the introduction of a refined complexity measure, whereas the inverse approximation theorem is new.
Optimization and generalization are two essential aspects of statistical machine learning. In this paper, we propose a framework to connect optimization with generalization by analyzing the generalization error based on the optimization trajectory under the gradient flow algorithm. The key ingredient of this framework is the Uniform-LGI, a property that is generally satisfied when training machine learning models. Leveraging the Uniform-LGI, we first derive convergence rates for gradient flow algorithm, then we give generalization bounds for a large class of machine learning models. We further apply our framework to three distinct machine learning models: linear regression, kernel regression, and two-layer neural networks. Through our approach, we obtain generalization estimates that match or extend previous results.
The invariant distribution, which is characterized by the stationary Fokker-Planck equation, is an important object in the study of randomly perturbed dynamical systems. Traditional numerical methods for computing the invariant distribution based on the Fokker-Planck equation, such as finite difference or finite element methods, are limited to low-dimensional systems due to the curse of dimensionality. In this work, we propose a deep learning based method to compute the generalized potential, i.e. the negative logarithm of the invariant distribution multiplied by the noise. The idea of the method is to learn a decomposition of the force field, as specified by the Fokker-Planck equation, from the trajectory data. The potential component of the decomposition gives the generalized potential. The method can deal with high-dimensional systems, possibly with partially known dynamics. Using the generalized potential also allows us to deal with systems at low temperatures, where the invariant distribution becomes singular around the metastable states. These advantages make it an efficient method to analyze invariant distributions for practical dynamical systems. The effectiveness of the proposed method is demonstrated by numerical examples.
We study the approximation properties of convolutional architectures applied to time series modelling, which can be formulated mathematically as a functional approximation problem. In the recurrent setting, recent results reveal an intricate connection between approximation efficiency and memory structures in the data generation process. In this paper, we derive parallel results for convolutional architectures, with WaveNet being a prime example. Our results reveal that in this new setting, approximation efficiency is not only characterised by memory, but also additional fine structures in the target relationship. This leads to a novel definition of spectrum-based regularity that measures the complexity of temporal relationships under the convolutional approximation scheme. These analyses provide a foundation to understand the differences between architectural choices for time series modelling and can give theoretically grounded guidance for practical applications.
We study the learning of numerical algorithms for scientific computing, which combines mathematically driven, handcrafted design of general algorithm structure with a data-driven adaptation to specific classes of tasks. This represents a departure from the classical approaches in numerical analysis, which typically do not feature such learning-based adaptations. As a case study, we develop a machine learning approach that automatically learns effective solvers for initial value problems in the form of ordinary differential equations (ODEs), based on the Runge-Kutta (RK) integrator architecture. We show that we can learn high-order integrators for targeted families of differential equations without the need for computing integrator coefficients by hand. Moreover, we demonstrate that in certain cases we can obtain superior performance to classical RK methods. This can be attributed to certain properties of the ODE families being identified and exploited by the approach. Overall, this work demonstrates an effective learning-based approach to the design of algorithms for the numerical solution of differential equations. This can be readily extended to other numerical tasks.
Tian Huang, Siong Thye Goh, Sabrish Gopalakrishnan et al.
An increasingly popular method for solving a constrained combinatorial optimisation problem is to first convert it into a quadratic unconstrained binary optimisation (QUBO) problem, and solve it using a standard QUBO solver. However, this relaxation introduces hyper-parameters that balance the objective and penalty terms for the constraints, and their chosen values significantly impact performance. Hence, tuning these parameters is an important problem. Existing generic hyper-parameter tuning methods require multiple expensive calls to a QUBO solver, making them impractical for performance critical applications when repeated solutions of similar combinatorial optimisation problems are required. In this paper, we propose the QROSS method, in which we build surrogate models of QUBO solvers via learning from solver data on a collection of instances of a problem. In this way, we are able capture the common structure of the instances and their interactions with the solver, and produce good choices of penalty parameters with fewer number of calls to the QUBO solver. We take the Traveling Salesman Problem (TSP) as a case study, where we demonstrate that our method can find better solutions with fewer calls to QUBO solver compared with conventional hyper-parameter tuning techniques. Moreover, with simple adaptation methods, QROSS is shown to generalise well to out-of-distribution datasets and different types of QUBO solvers.
Despite their success in massive engineering applications, deep neural networks are vulnerable to various perturbations due to their black-box nature. Recent study has shown that a deep neural network can misclassify the data even if the input data is perturbed by an imperceptible amount. In this paper, we address the robustness issue of neural networks by a novel close-loop control method from the perspective of dynamic systems. Instead of modifying the parameters in a fixed neural network architecture, a close-loop control process is added to generate control signals adaptively for the perturbed or corrupted data. We connect the robustness of neural networks with optimal control using the geometrical information of underlying data to design the control objective. The detailed analysis shows how the embedding manifolds of state trajectory affect error estimation of the proposed method. Our approach can simultaneously maintain the performance on clean data and improve the robustness against many types of data perturbations. It can also further improve the performance of robustly trained neural networks against different perturbations. To the best of our knowledge, this is the first work that improves the robustness of neural networks with close-loop control.
Despite the empirical success in various domains, it has been revealed that deep neural networks are vulnerable to maliciously perturbed input data that much degrade their performance. This is known as adversarial attacks. To counter adversarial attacks, adversarial training formulated as a form of robust optimization has been demonstrated to be effective. However, conducting adversarial training brings much computational overhead compared with standard training. In order to reduce the computational cost, we propose an annealing mechanism, Amata, to reduce the overhead associated with adversarial training. The proposed Amata is provably convergent, well-motivated from the lens of optimal control theory and can be combined with existing acceleration methods to further enhance performance. It is demonstrated that on standard datasets, Amata can achieve similar or better robustness with around 1/3 to 1/2 the computational time compared with traditional methods. In addition, Amata can be incorporated into other adversarial training acceleration algorithms (e.g. YOPO, Free, Fast, and ATTA), which leads to further reduction in computational time on large-scale problems.
The quasipotential is a natural generalization of the concept of energy functions to non-equilibrium systems. In the analysis of rare events in stochastic dynamics, it plays a central role in characterizing the statistics of transition events and the likely transition paths. However, computing the quasipotential is challenging, especially in high dimensional dynamical systems where a global landscape is sought. Traditional methods based on the dynamic programming principle or path space minimization tend to suffer from the curse of dimensionality. In this paper, we propose a simple and efficient machine learning method to resolve this problem. The key idea is to learn an orthogonal decomposition of the vector field that drives the dynamics, from which one can identify the quasipotential. We demonstrate on various example systems that our method can effectively compute quasipotential landscapes without requiring spatial discretization or solving path-space optimization problems. Moreover, the method is purely data driven in the sense that only observed trajectories of the dynamics are required for the computation of the quasipotential. These properties make it a promising method to enable the general application of quasipotential analysis to dynamical systems away from equilibrium.
The future of mobility-as-a-Service (Maas)should embrace an integrated system of ride-hailing, street-hailing and ride-sharing with optimised intelligent vehicle routing in response to a real-time, stochastic demand pattern. We aim to optimise routing policies for a large fleet of vehicles for street-hailing services, given a stochastic demand pattern in small to medium-sized road networks. A model-based dispatch algorithm, a high performance model-free reinforcement learning based algorithm and a novel hybrid algorithm combining the benefits of both the top-down approach and the model-free reinforcement learning have been proposed to route the \emph{vacant} vehicles. We design our reinforcement learning based routing algorithm using proximal policy optimisation and combined intrinsic and extrinsic rewards to strike a balance between exploration and exploitation. Using a large-scale agent-based microscopic simulation platform to evaluate our proposed algorithms, our model-free reinforcement learning and hybrid algorithm show excellent performance on both artificial road network and community-based Singapore road network with empirical demands, and our hybrid algorithm can significantly accelerate the model-free learner in the process of learning.
We study the approximation properties and optimization dynamics of recurrent neural networks (RNNs) when applied to learn input-output relationships in temporal data. We consider the simple but representative setting of using continuous-time linear RNNs to learn from data generated by linear relationships. Mathematically, the latter can be understood as a sequence of linear functionals. We prove a universal approximation theorem of such linear functionals, and characterize the approximation rate and its relation with memory. Moreover, we perform a fine-grained dynamical analysis of training linear RNNs, which further reveal the intricate interactions between memory and learning. A unifying theme uncovered is the non-trivial effect of memory, a notion that can be made precise in our framework, on approximation and optimization: when there is long term memory in the target, it takes a large number of neurons to approximate it. Moreover, the training process will suffer from slow downs. In particular, both of these effects become exponentially more pronounced with memory - a phenomenon we call the "curse of memory". These analyses represent a basic step towards a concrete mathematical understanding of new phenomenon that may arise in learning temporal relationships using recurrent architectures.
We propose a systematic method for learning stable and physically interpretable dynamical models using sampled trajectory data from physical processes based on a generalized Onsager principle. The learned dynamics are autonomous ordinary differential equations parameterized by neural networks that retain clear physical structure information, such as free energy, diffusion, conservative motion and external forces. For high dimensional problems with a low dimensional slow manifold, an autoencoder with metric preserving regularization is introduced to find the low dimensional generalized coordinates on which we learn the generalized Onsager dynamics. Our method exhibits clear advantages over existing methods on benchmark problems for learning ordinary differential equations. We further apply this method to study Rayleigh-Benard convection and learn Lorenz-like low dimensional autonomous reduced order models that capture both qualitative and quantitative properties of the underlying dynamics. This forms a general approach to building reduced order models for forced dissipative systems.
Zekun Ren, Siyu Isaac Parker Tian, Juhwan Noh et al.
Realizing general inverse design could greatly accelerate the discovery of new materials with user-defined properties. However, state-of-the-art generative models tend to be limited to a specific composition or crystal structure. Herein, we present a framework capable of general inverse design (not limited to a given set of elements or crystal structures), featuring a generalized invertible representation that encodes crystals in both real and reciprocal space, and a property-structured latent space from a variational autoencoder (VAE). In three design cases, the framework generates 142 new crystals with user-defined formation energies, bandgap, thermoelectric (TE) power factor, and combinations thereof. These generated crystals, absent in the training database, are validated by first-principles calculations. The success rates (number of first-principles-validated target-satisfying crystals/number of designed crystals) ranges between 7.1% and 38.9%. These results represent a significant step toward property-driven general inverse design using generative models, although practical challenges remain when coupled with experimental synthesis.
We present the viewpoint that optimization problems encountered in machine learning can often be interpreted as minimizing a convex functional over a function space, but with a non-convex constraint set introduced by model parameterization. This observation allows us to repose such problems via a suitable relaxation as convex optimization problems in the space of distributions over the training parameters. We derive some simple relationships between the distribution-space problem and the original problem, e.g. a distribution-space solution is at least as good as a solution in the original space. Moreover, we develop a numerical algorithm based on mixture distributions to perform approximate optimization directly in distribution space. Consistency of this approximation is established and the numerical efficacy of the proposed algorithm is illustrated on simple examples. In both theory and practice, this formulation provides an alternative approach to large-scale optimization in machine learning.
While current machine learning models have impressive performance over a wide range of applications, their large size and complexity render them unsuitable for tasks such as remote monitoring on edge devices with limited storage and computational power. A naive approach to resolve this on the model level is to use simpler architectures, but this sacrifices prediction accuracy and is unsuitable for monitoring applications requiring accurate detection of the onset of adverse events. In this paper, we propose an alternative solution to this problem by decomposing the predictive model as the sum of a simple function which serves as a local monitoring tool, and a complex correction term to be evaluated on the server. A sign requirement is imposed on the latter to ensure that the local monitoring function is safe, in the sense that it can effectively serve as an early warning system. Our analysis quantifies the trade-offs between model complexity and performance, and serves as a guidance for architecture design. We validate our proposed framework on a series of monitoring experiments, where we succeed at learning monitoring models with significantly reduced complexity that minimally violate the safety requirement. More broadly, our framework is useful for learning classifiers in applications where false negatives are significantly more costly compared to false positives.
We build on the dynamical systems approach to deep learning, where deep residual networks are idealized as continuous-time dynamical systems, from the approximation perspective. In particular, we establish general sufficient conditions for universal approximation using continuous-time deep residual networks, which can also be understood as approximation theories in $L^p$ using flow maps of dynamical systems. In specific cases, rates of approximation in terms of the time horizon are also established. Overall, these results reveal that composition function approximation through flow maps present a new paradigm in approximation theory and contributes to building a useful mathematical framework to investigate deep learning.
Distributed optimization often consists of two updating phases: local optimization and inter-node communication. Conventional approaches require working nodes to communicate with the server every one or few iterations to guarantee convergence. In this paper, we establish a completely different conclusion that each node can perform an arbitrary number of local optimization steps before communication. Moreover, we show that the more local updating can reduce the overall communication, even for an infinity number of steps where each node is free to update its local model to near-optimality before exchanging information. The extra assumption we make is that the optimal sets of local loss functions have a non-empty intersection, which is inspired by the over-paramterization phenomenon in large-scale optimization and deep learning. Our theoretical findings are confirmed by both distributed convex optimization and deep learning experiments.
Jatin N. Kumar, Qianxiao Li, Karen Y. T. Tang et al.
Inverse design is an outstanding challenge in disordered systems with multiple length scales such as polymers, particularly when designing polymers with desired phase behavior. We demonstrate high-accuracy tuning of poly(2-oxazoline) cloud point via machine learning. With a design space of four repeating units and a range of molecular masses, we achieve an accuracy of 4 °C root mean squared error (RMSE) in a temperature range of 24-90 °C, employing gradient boosting with decision trees. The RMSE is >3x better than linear and polynomial regression. We perform inverse design via particle-swarm optimization, predicting and synthesizing 17 polymers with constrained design at 4 target cloud points from 37 to 80 °C. Our approach challenges the status quo in polymer design with a machine learning algorithm, that is capable of fast and systematic discovery of new polymers.
We develop the mathematical foundations of the stochastic modified equations (SME) framework for analyzing the dynamics of stochastic gradient algorithms, where the latter is approximated by a class of stochastic differential equations with small noise parameters. We prove that this approximation can be understood mathematically as an weak approximation, which leads to a number of precise and useful results on the approximations of stochastic gradient descent (SGD), momentum SGD and stochastic Nesterov's accelerated gradient method in the general setting of stochastic objectives. We also demonstrate through explicit calculations that this continuous-time approach can uncover important analytical insights into the stochastic gradient algorithms under consideration that may not be easy to obtain in a purely discrete-time setting.
Despite its empirical success and recent theoretical progress, there generally lacks a quantitative analysis of the effect of batch normalization (BN) on the convergence and stability of gradient descent. In this paper, we provide such an analysis on the simple problem of ordinary least squares (OLS). Since precise dynamical properties of gradient descent (GD) is completely known for the OLS problem, it allows us to isolate and compare the additional effects of BN. More precisely, we show that unlike GD, gradient descent with BN (BNGD) converges for arbitrary learning rates for the weights, and the convergence remains linear under mild conditions. Moreover, we quantify two different sources of acceleration of BNGD over GD -- one due to over-parameterization which improves the effective condition number and another due having a large range of learning rates giving rise to fast descent. These phenomena set BNGD apart from GD and could account for much of its robustness properties. These findings are confirmed quantitatively by numerical experiments, which further show that many of the uncovered properties of BNGD in OLS are also observed qualitatively in more complex supervised learning problems.
Deep learning is formulated as a discrete-time optimal control problem. This allows one to characterize necessary conditions for optimality and develop training algorithms that do not rely on gradients with respect to the trainable parameters. In particular, we introduce the discrete-time method of successive approximations (MSA), which is based on the Pontryagin's maximum principle, for training neural networks. A rigorous error estimate for the discrete MSA is obtained, which sheds light on its dynamics and the means to stabilize the algorithm. The developed methods are applied to train, in a rather principled way, neural networks with weights that are constrained to take values in a discrete set. We obtain competitive performance and interestingly, very sparse weights in the case of ternary networks, which may be useful in model deployment in low-memory devices.
The continuous dynamical system approach to deep learning is explored in order to devise alternative frameworks for training algorithms. Training is recast as a control problem and this allows us to formulate necessary optimality conditions in continuous time using the Pontryagin's maximum principle (PMP). A modification of the method of successive approximations is then used to solve the PMP, giving rise to an alternative training algorithm for deep learning. This approach has the advantage that rigorous error estimates and convergence results can be established. We also show that it may avoid some pitfalls of gradient-based methods, such as slow convergence on flat landscapes near saddle points. Furthermore, we demonstrate that it obtains favorable initial convergence rate per-iteration, provided Hamiltonian maximization can be efficiently carried out - a step which is still in need of improvement. Overall, the approach opens up new avenues to attack problems associated with deep learning, such as trapping in slow manifolds and inapplicability of gradient-based methods for discrete trainable variables.
We develop the method of stochastic modified equations (SME), in which stochastic gradient algorithms are approximated in the weak sense by continuous-time stochastic differential equations. We exploit the continuous formulation together with optimal control theory to derive novel adaptive hyper-parameter adjustment policies. Our algorithms have competitive performance with the added benefit of being robust to varying models and datasets. This provides a general methodology for the analysis and design of stochastic gradient algorithms.