We address the problem of learning the dynamics of an unknown non-parametric system linking a target and a feature time series. The feature time series is measured on a sparse and irregular grid, while we have access to only a few points of the target time series. Once learned, we can use these dynamics to predict values of the target from the previous values of the feature time series. We frame this task as learning the solution map of a controlled differential equation (CDE). By leveraging the rich theory of signatures, we are able to cast this non-linear problem as a high-dimensional linear regression. We provide an oracle bound on the prediction error which exhibits explicit dependencies on the individual-specific sampling schemes. Our theoretical results are illustrated by simulations which show that our method outperforms existing algorithms for recovering the full time series while being computationally cheap. We conclude by demonstrating its potential on real-world epidemiological data.
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We study the offline contextual bandit problem, where we aim to acquire an optimal policy using observational data. However, this data usually contains two deficiencies: (i) some variables that confound actions are not observed, and (ii) missing observations exist in the collected data. Unobserved confounders lead to a confounding bias and missing observations cause bias and inefficiency problems. To overcome these challenges and learn the optimal policy from the observed dataset, we present a new algorithm called Causal-Adjusted Pessimistic (CAP) policy learning, which forms the reward function as the solution of an integral equation system, builds a confidence set, and greedily takes action with pessimism. With mild assumptions on the data, we develop an upper bound to the suboptimality of CAP for the offline contextual bandit problem.
We develop an optimization-based algorithm for parametric model order reduction (PMOR) of linear time-invariant dynamical systems. Our method aims at minimizing the $\mathcal{H}_\infty \otimes \mathcal{L}_\infty$ approximation error in the frequency and parameter domain by an optimization of the reduced order model (ROM) matrices. State-of-the-art PMOR methods often compute several nonparametric ROMs for different parameter samples, which are then combined to a single parametric ROM. However, these parametric ROMs can have a low accuracy between the utilized sample points. In contrast, our optimization-based PMOR method minimizes the approximation error across the entire parameter domain. Moreover, due to our flexible approach of optimizing the system matrices directly, we can enforce favorable features such as a port-Hamiltonian structure in our ROMs across the entire parameter domain. Our method is an extension of the recently developed SOBMOR-algorithm to parametric systems. We extend both the ROM parameterization and the adaptive sampling procedure to the parametric case. Several numerical examples demonstrate the effectiveness and high accuracy of our method in a comparison with other PMOR methods.
It is a common phenomenon that for high-dimensional and nonparametric statistical models, rate-optimal estimators balance squared bias and variance. Although this balancing is widely observed, little is known whether methods exist that could avoid the trade-off between bias and variance. We propose a general strategy to obtain lower bounds on the variance of any estimator with bias smaller than a prespecified bound. This shows to which extent the bias-variance trade-off is unavoidable and allows to quantify the loss of performance for methods that do not obey it. The approach is based on a number of abstract lower bounds for the variance involving the change of expectation with respect to different probability measures as well as information measures such as the Kullback-Leibler or $\chi^2$-divergence. In a second part of the article, the abstract lower bounds are applied to several statistical models including the Gaussian white noise model, a boundary estimation problem, the Gaussian sequence model and the high-dimensional linear regression model. For these specific statistical applications, different types of bias-variance trade-offs occur that vary considerably in their strength. For the trade-off between integrated squared bias and integrated variance in the Gaussian white noise model, we propose to combine the general strategy for lower bounds with a reduction technique. This allows us to reduce the original problem to a lower bound on the bias-variance trade-off for estimators with additional symmetry properties in a simpler statistical model. In the Gaussian sequence model, different phase transitions of the bias-variance trade-off occur. Although there is a non-trivial interplay between bias and variance, the rate of the squared bias and the variance do not have to be balanced in order to achieve the minimax estimation rate.
Numerous physics theories are rooted in partial differential equations (PDEs). However, the increasingly intricate physics equations, especially those that lack analytic solutions or closed forms, have impeded the further development of physics. Computationally solving PDEs by classic numerical approaches suffers from the trade-off between accuracy and efficiency and is not applicable to the empirical data generated by unknown latent PDEs. To overcome this challenge, we present KoopmanLab, an efficient module of the Koopman neural operator family, for learning PDEs without analytic solutions or closed forms. Our module consists of multiple variants of the Koopman neural operator (KNO), a kind of mesh-independent neural-network-based PDE solvers developed following dynamic system theory. The compact variants of KNO can accurately solve PDEs with small model sizes while the large variants of KNO are more competitive in predicting highly complicated dynamic systems govern by unknown, high-dimensional, and non-linear PDEs. All variants are validated by mesh-independent and long-term prediction experiments implemented on representative PDEs (e.g., the Navier-Stokes equation and the Bateman-Burgers equation in fluid mechanics) and ERA5 (i.e., one of the largest high-resolution global-scale climate data sets in earth physics). These demonstrations suggest the potential of KoopmanLab to be a fundamental tool in diverse physics studies related to equations or dynamic systems.
Safety is critical in robotic tasks. Energy function based methods have been introduced to address the problem. To ensure safety in the presence of control limits, we need to design an energy function that results in persistently feasible safe control at all system states. However, designing such an energy function for high-dimensional nonlinear systems remains challenging. Considering the fact that there are redundant dynamics in high dimensional systems with respect to the safety specifications, this paper proposes a novel approach called abstract safe control. We propose a system abstraction method that enables the design of energy functions on a low-dimensional model. Then we can synthesize the energy function with respect to the low-dimensional model to ensure persistent feasibility. The resulting safe controller can be directly transferred to other systems with the same abstraction, e.g., when a robot arm holds different tools. The proposed approach is demonstrated on a 7-DoF robot arm (14 states) both in simulation and real-world. Our method always finds feasible control and achieves zero safety violations in 500 trials on 5 different systems.
The dynamics of the power system are described by a system of differential-algebraic equations. Time-domain simulations are used to understand the evolution of the system dynamics. These simulations can be computationally expensive due to the stiffness of the system which requires the use of finely discretized time-steps. By increasing the allowable time-step size, we aim to accelerate such simulations. In this paper, we use the observation that even though the individual components are described using both algebraic and differential equations, their coupling only involves algebraic equations. Following this observation, we use Neural Networks (NNs) to approximate the components' state evolution, leading to fast, accurate, and numerically stable approximators, which enable larger time-steps. To account for effects of the network on the components and vice-versa, the NNs take the temporal evolution of the coupling algebraic variables as an input for their prediction. We initially estimate this temporal evolution and then update it in an iterative fashion using the Newton-Raphson algorithm. The involved Jacobian matrix is calculated with Automatic Differentiation and its size depends only on the network size but not on the component dynamics. We demonstrate this NN-based simulator on the IEEE 9-bus test case with 3 generators.
Many complex engineering systems can be represented in a topological form, such as graphs. This paper utilizes a machine learning technique called Geometric Deep Learning (GDL) to aid designers with challenging, graph-centric design problems. The strategy presented here is to take the graph data and apply GDL to seek the best realizable performing solution effectively and efficiently with lower computational costs. This case study used here is the synthesis of analog electrical circuits that attempt to match a specific frequency response within a particular frequency range. Previous studies utilized an enumeration technique to generate 43,249 unique undirected graphs presenting valid potential circuits. Unfortunately, determining the sizing and performance of many circuits can be too expensive. To reduce computational costs with a quantified trade-off in accuracy, the fraction of the circuit graphs and their performance are used as input data to a classification-focused GDL model. Then, the GDL model can be used to predict the remainder cheaply, thus, aiding decision-makers in the search for the best graph solutions. The results discussed in this paper show that additional graph-based features are useful, favorable total set classification accuracy of 80\% in using only 10\% of the graphs, and iteratively-built GDL models can further subdivide the graphs into targeted groups with medians significantly closer to the best graph and containing 87 of the top 100 best performing graphs.
We present DiffXPBD, a novel and efficient analytical formulation for the differentiable position-based simulation of compliant constrained dynamics (XPBD). Our proposed method allows computation of gradients of numerous parameters with respect to a goal function simultaneously leveraging a performant simulation model. The method is efficient, thus enabling differentiable simulations of high resolution geometries and degrees of freedom (DoFs). Collisions are naturally included in the framework. Our differentiable model allows a user to easily add additional optimization variables. Every control variable gradient requires the computation of only a few partial derivatives which can be computed using automatic differentiation code. We demonstrate the efficacy of the method with examples such as elastic material parameter estimation, initial value optimization, optimizing for underlying body shape and pose by only observing the clothing, and optimizing a time-varying external force sequence to match sparse keyframe shapes at specific times. Our approach demonstrates excellent efficiency and we demonstrate this on high resolution meshes with optimizations involving over 26 million degrees of freedom. Making an existing solver differentiable requires only a few modifications and the model is compatible with both modern CPU and GPU multi-core hardware.
This manuscript portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.
To address the sparsity and cold start problem of collaborative filtering, researchers usually make use of side information, such as social networks or item attributes, to improve recommendation performance. This paper considers the knowledge graph as the source of side information. To address the limitations of existing embedding-based and path-based methods for knowledge-graph-aware recommendation, we propose Ripple Network, an end-to-end framework that naturally incorporates the knowledge graph into recommender systems. Similar to actual ripples propagating on the surface of water, Ripple Network stimulates the propagation of user preferences over the set of knowledge entities by automatically and iteratively extending a user's potential interests along links in the knowledge graph. The multiple "ripples" activated by a user's historically clicked items are thus superposed to form the preference distribution of the user with respect to a candidate item, which could be used for predicting the final clicking probability. Through extensive experiments on real-world datasets, we demonstrate that Ripple Network achieves substantial gains in a variety of scenarios, including movie, book and news recommendation, over several state-of-the-art baselines.
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