Search ResultsShowing 1-11 of 11
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arXiv:2401.10385 (Published 2024-01-18)
Approximation of Solution Operators for High-dimensional PDEs
Comments: 14 pages, 4 page appendix, 4 figuresWe propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural network, we connect the evolution of the model parameters with trajectories in a corresponding function space. Using the computational technique of neural ordinary differential equation, we learn the control over the parameter space such that from any initial starting point, the controlled trajectories closely approximate the solutions to the PDE. Approximation accuracy is justified for a general class of second-order nonlinear PDEs. Numerical results are presented for several high-dimensional PDEs, including real-world applications to solving Hamilton-Jacobi-Bellman equations. These are demonstrated to show the accuracy and efficiency of the proposed method.
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arXiv:2311.05738 (Published 2023-11-09)
On Optimal Control at the Onset of a New Viral Outbreak
Categories: math.OCAn optimal control problem for the early stage of an infectious disease outbreak is considered. At that stage, control is often limited to non-medical interventions like social distancing and other behavioral changes. We show that the running cost of control satisfying mild, problem-specific, conditions generates an optimal control strategy that stays inside its admissible set for the entire duration of the study period $[0 ,T]$. For the optimal control problem, restricted by SIR compartmental model of disease transmission, we prove that the optimal control strategy, $u(t)$, may be growing until some moment $\bar{t} \in [0 ,T)$. However, for any $t \in [\bar{t}, T]$, the function $u(t)$ will decline as $t$ approaches $T$, which may cause the number of newly infected people to increase. So, the window from $0$ to $\bar{t}$ is the time for public health officials to prepare alternative mitigation measures, such as vaccines, testing, antiviral medications, and others. Our theoretical findings are illustrated with numerical examples showing optimal control strategies for various cost functions and weights. Simulation results provide a comprehensive demonstration of the effects of control on the epidemic spread and mitigation expenses, which can serve as invaluable references for public health officials.
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arXiv:2307.13135 (Published 2023-07-24)
High-dimensional Optimal Density Control with Wasserstein Metric Matching
Comments: 8 pages, 4 figures. Accepted for IEEE Conference on Decision and Control 2023Categories: math.OCKeywords: high-dimensional optimal density control, wasserstein metric matching, deep network, target distribution, fast automatic differentiationTags: conference paperWe present a novel computational framework for density control in high-dimensional state spaces. The considered dynamical system consists of a large number of indistinguishable agents whose behaviors can be collectively modeled as a time-evolving probability distribution. The goal is to steer the agents from an initial distribution to reach (or approximate) a given target distribution within a fixed time horizon at minimum cost. To tackle this problem, we propose to model the drift as a nonlinear reduced-order model, such as a deep network, and enforce the matching to the target distribution at terminal time either strictly or approximately using the Wasserstein metric. The resulting saddle-point problem can be solved by an effective numerical algorithm that leverages the excellent representation power of deep networks and fast automatic differentiation for this challenging high-dimensional control problem. A variety of numerical experiments were conducted to demonstrate the performance of our method.
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arXiv:2302.00045 (Published 2023-01-31)
Neural Control of Parametric Solutions for High-dimensional Evolution PDEs
Comments: 23 pagesWe develop a novel computational framework to approximate solution operators of evolution partial differential equations (PDEs). By employing a general nonlinear reduced-order model, such as a deep neural network, to approximate the solution of a given PDE, we realize that the evolution of the model parameter is a control problem in the parameter space. Based on this observation, we propose to approximate the solution operator of the PDE by learning the control vector field in the parameter space. From any initial value, this control field can steer the parameter to generate a trajectory such that the corresponding reduced-order model solves the PDE. This allows for substantially reduced computational cost to solve the evolution PDE with arbitrary initial conditions. We also develop comprehensive error analysis for the proposed method when solving a large class of semilinear parabolic PDEs. Numerical experiments on different high-dimensional evolution PDEs with various initial conditions demonstrate the promising results of the proposed method.
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arXiv:2204.03804 (Published 2022-04-08)
A Learnable Variational Model for Joint Multimodal MRI Reconstruction and Synthesis
Comments: 12 pagesGenerating multi-contrasts/modal MRI of the same anatomy enriches diagnostic information but is limited in practice due to excessive data acquisition time. In this paper, we propose a novel deep-learning model for joint reconstruction and synthesis of multi-modal MRI using incomplete k-space data of several source modalities as inputs. The output of our model includes reconstructed images of the source modalities and high-quality image synthesized in the target modality. Our proposed model is formulated as a variational problem that leverages several learnable modality-specific feature extractors and a multimodal synthesis module. We propose a learnable optimization algorithm to solve this model, which induces a multi-phase network whose parameters can be trained using multi-modal MRI data. Moreover, a bilevel-optimization framework is employed for robust parameter training. We demonstrate the effectiveness of our approach using extensive numerical experiments.
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Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation
Comments: Accepted to 61st IEEE Conference on Decision and ControlKeywords: lyapunov function approximation, versatile deep neural network architecture, approximate lyapunov functions, dynamical systems, lyapunov-net guarantees positive definitenessTags: conference paperWe develop a versatile deep neural network architecture, called Lyapunov-Net, to approximate Lyapunov functions of dynamical systems in high dimensions. Lyapunov-Net guarantees positive definiteness, and thus it can be easily trained to satisfy the negative orbital derivative condition, which only renders a single term in the empirical risk function in practice. This significantly reduces the number of hyper-parameters compared to existing methods. We also provide theoretical justifications on the approximation power of Lyapunov-Net and its complexity bounds. We demonstrate the efficiency of the proposed method on nonlinear dynamical systems involving up to 30-dimensional state spaces, and show that the proposed approach significantly outperforms the state-of-the-art methods.
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arXiv:2109.09738 (Published 2021-09-20)
An Optimal Control Framework for Joint-channel Parallel MRI Reconstruction without Coil Sensitivities
Comments: 13 pagesGoal: This work aims at developing a novel calibration-free fast parallel MRI (pMRI) reconstruction method incorporate with discrete-time optimal control framework. The reconstruction model is designed to learn a regularization that combines channels and extracts features by leveraging the information sharing among channels of multi-coil images. We propose to recover both magnitude and phase information by taking advantage of structured multiplayer convolutional networks in image and Fourier spaces. Methods: We develop a novel variational model with a learnable objective function that integrates an adaptive multi-coil image combination operator and effective image regularization in the image and Fourier spaces. We cast the reconstruction network as a structured discrete-time optimal control system, resulting in an optimal control formulation of parameter training where the parameters of the objective function play the role of control variables. We demonstrate that the Lagrangian method for solving the control problem is equivalent to back-propagation, ensuring the local convergence of the training algorithm. Results: We conduct a large number of numerical experiments of the proposed method with comparisons to several state-of-the-art pMRI reconstruction networks on real pMRI datasets. The numerical results demonstrate the promising performance of the proposed method evidently. Conclusion: The proposed method provides a general deep network design and training framework for efficient joint-channel pMRI reconstruction. Significance: By learning multi-coil image combination operator and performing regularizations in both image domain and k-space domain, the proposed method achieves a highly efficient image reconstruction network for pMRI.
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arXiv:2106.02608 (Published 2021-06-03)
Influence Estimation and Maximization via Neural Mean-Field Dynamics
Comments: 26 pages, 6 figures. arXiv admin note: text overlap with arXiv:2006.09449We propose a novel learning framework using neural mean-field (NMF) dynamics for inference and estimation problems on heterogeneous diffusion networks. Our new framework leverages the Mori-Zwanzig formalism to obtain an exact evolution equation of the individual node infection probabilities, which renders a delay differential equation with memory integral approximated by learnable time convolution operators. Directly using information diffusion cascade data, our framework can simultaneously learn the structure of the diffusion network and the evolution of node infection probabilities. Connections between parameter learning and optimal control are also established, leading to a rigorous and implementable algorithm for training NMF. Moreover, we show that the projected gradient descent method can be employed to solve the challenging influence maximization problem, where the gradient is computed extremely fast by integrating NMF forward in time just once in each iteration. Extensive empirical studies show that our approach is versatile and robust to variations of the underlying diffusion network models, and significantly outperform existing approaches in accuracy and efficiency on both synthetic and real-world data.
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arXiv:2003.06748 (Published 2020-03-15)
A Novel Learnable Gradient Descent Type Algorithm for Non-convex Non-smooth Inverse Problems
Optimization algorithms for solving nonconvex inverse problem have attracted significant interests recently. However, existing methods require the nonconvex regularization to be smooth or simple to ensure convergence. In this paper, we propose a novel gradient descent type algorithm, by leveraging the idea of residual learning and Nesterov's smoothing technique, to solve inverse problems consisting of general nonconvex and nonsmooth regularization with provable convergence. Moreover, we develop a neural network architecture intimating this algorithm to learn the nonlinear sparsity transformation adaptively from training data, which also inherits the convergence to accommodate the general nonconvex structure of this learned transformation. Numerical results demonstrate that the proposed network outperforms the state-of-the-art methods on a variety of different image reconstruction problems in terms of efficiency and accuracy.
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arXiv:1604.05649 (Published 2016-04-19)
Decentralized Consensus Algorithm with Delayed and Stochastic Gradients
Categories: math.OCWe analyze the convergence of a decentralized consensus algorithm with delayed gradient information across the network. The nodes in the network privately hold parts of the objective function and collaboratively solve for the consensus optimal solution of the total objective while they can only communicate with their immediate neighbors. In real-world networks, it is often difficult and sometimes impossible to synchronize the nodes, and therefore they have to use stale gradient information during computations. We show that, as long as the random delays are bounded in expectation and a proper diminishing step size policy is employed, the iterates generated by decentralized gradient descent method converge to a consensual optimal solution. Convergence rates of both objective and consensus are derived. Numerical results on a number of synthetic problems are presented to show the performance of the method.
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Projection Onto A Simplex
Categories: math.OCThis mini-paper presents a fast and simple algorithm to compute the projection onto the canonical simplex $\triangle^n$. Utilizing the Moreau's identity, we show that the problem is essentially a univariate minimization and the objective function is strictly convex and continuously differentiable. Moreover, it is shown that there are at most n candidates which can be computed explicitly, and the minimizer is the only one that falls into the correct interval.