arXiv subjects

Paula Chen

Publications and source records attributed to Paula Chen.

8 recordsLinked to original sources

arXiv2026

PI-SONet: A Physics-Informed Symplectic Operator Network for Real-Time Optimal Control of Multi-Agent Systems

Many real-life applications involve controlling high-dimensional multi-agent systems in real-time. Existing optimal control solvers often suffer from the curse-of-dimensionality and require complete rerunning for each new problem setting. We target nonconvex, nonlinear problems in 100s of dimensions by introducing PI-SONet (Physics-Informed Symplectic Operator Network), a structure-preserving operator learning framework for solving parameterized families of optimal control problems and their Pontraygin Maximum Principle (PMP) systems. PI-SONet combines a latent right-space solver with a conditional symplectic operator to produce tractable Hamiltonian trajectories in a computationally efficient auxiliary space and transform them back to physical space. This decomposition yields a \textit{single} trained operator that approximates the PMP solution map, inherently preserves Hamiltonian structure, and generalizes across unseen problem configurations. Unlike existing methods, which are fundamentally single-instance solvers, PI-SONet achieves sub-second inferences on new problem instances, equating to up to 10,000x speedup over representative baselines. These results suggest that structure-preserving neural operators provide a practical route toward reusable, real-time surrogates for high-dimensional optimal control.

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arXiv2026

Algorithms and Differential Game Representations for Exploring Nonconvex Pareto Fronts in High Dimensions

We develop a new Hamiton-Jacobi (HJ) and differential game approach for exploring the Pareto front of (constrained) multi-objective optimization (MOO) problems. Given a preference function, we embed the scalarized MOO problem into the value function of a parameterized zero-sum game, whose upper value solves a first-order HJ equation that admits a Hopf-Lax representation formula. For each parameter value, this representation yields an inner minimizer that can be interpreted as an approximate solution to a shifted scalarization of the original MOO problem. Under mild assumptions, the resulting family of solutions maps to a dense subset of the weak Pareto front. Finally, we propose a primal-dual algorithm based on this approach for solving the corresponding optimality system. Numerical experiments show that our algorithm mitigates the curse of dimensionality (scaling polynomially with the dimension of the decision and objective spaces) and is able to expose continuous curves along nonconvex Pareto fronts in 100D in just $\sim$100 seconds.

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arXiv2025

A Model-Free Data-Driven Algorithm for Continuous-Time Control

Presented is an algorithm to synthesize an infinite-horizon LQR optimal feedback controller for continuous-time systems. The algorithm does not require knowledge of the system dynamics, but instead uses only a finite-length sampling of (possibly suboptimal) input-output data. The algorithm is based on a constrained optimization problem that enforces a necessary condition on the dynamics of the optimal value function along an arbitrary trajectory. This paper presents the derivation as well as shows examples applied to both linear and nonlinear systems inspired by air vehicles.

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arXiv2024

Leveraging viscous Hamilton-Jacobi PDEs for uncertainty quantification in scientific machine learning

Uncertainty quantification (UQ) in scientific machine learning (SciML) combines the powerful predictive power of SciML with methods for quantifying the reliability of the learned models. However, two major challenges remain: limited interpretability and expensive training procedures. We provide a new interpretation for UQ problems by establishing a new theoretical connection between some Bayesian inference problems arising in SciML and viscous Hamilton-Jacobi partial differential equations (HJ PDEs). Namely, we show that the posterior mean and covariance can be recovered from the spatial gradient and Hessian of the solution to a viscous HJ PDE. As a first exploration of this connection, we specialize to Bayesian inference problems with linear models, Gaussian likelihoods, and Gaussian priors. In this case, the associated viscous HJ PDEs can be solved using Riccati ODEs, and we develop a new Riccati-based methodology that provides computational advantages when continuously updating the model predictions. Specifically, our Riccati-based approach can efficiently add or remove data points to the training set invariant to the order of the data and continuously tune hyperparameters. Moreover, neither update requires retraining on or access to previously incorporated data. We provide several examples from SciML involving noisy data and \textit{epistemic uncertainty} to illustrate the potential advantages of our approach. In particular, this approach's amenability to data streaming applications demonstrates its potential for real-time inferences, which, in turn, allows for applications in which the predicted uncertainty is used to dynamically alter the learning process.

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arXiv2023

Leveraging Hamilton-Jacobi PDEs with time-dependent Hamiltonians for continual scientific machine learning

We address two major challenges in scientific machine learning (SciML): interpretability and computational efficiency. We increase the interpretability of certain learning processes by establishing a new theoretical connection between optimization problems arising from SciML and a generalized Hopf formula, which represents the viscosity solution to a Hamilton-Jacobi partial differential equation (HJ PDE) with time-dependent Hamiltonian. Namely, we show that when we solve certain regularized learning problems with integral-type losses, we actually solve an optimal control problem and its associated HJ PDE with time-dependent Hamiltonian. This connection allows us to reinterpret incremental updates to learned models as the evolution of an associated HJ PDE and optimal control problem in time, where all of the previous information is intrinsically encoded in the solution to the HJ PDE. As a result, existing HJ PDE solvers and optimal control algorithms can be reused to design new efficient training approaches for SciML that naturally coincide with the continual learning framework, while avoiding catastrophic forgetting. As a first exploration of this connection, we consider the special case of linear regression and leverage our connection to develop a new Riccati-based methodology for solving these learning problems that is amenable to continual learning applications. We also provide some corresponding numerical examples that demonstrate the potential computational and memory advantages our Riccati-based approach can provide.

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arXiv2023

Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems

Two key challenges in optimal control include efficiently solving high-dimensional problems and handling optimal control problems with state-dependent running costs. In this paper, we consider a class of optimal control problems whose running costs consist of a quadratic on the control variable and a convex, non-negative, piecewise affine function on the state variable. We provide the analytical solution for this class of optimal control problems as well as a Hopf-type representation formula for the corresponding Hamilton-Jacobi partial differential equations. Finally, we propose efficient numerical algorithms based on our Hopf-type representation formula, convex optimization algorithms, and min-plus techniques. We present several high-dimensional numerical examples, which demonstrate that our algorithms overcome the curse of dimensionality. We also describe a field-programmable gate array (FPGA) implementation of our numerical solver whose latency scales linearly in the spatial dimension and that achieves approximately a 40 times speedup compared to a parallelized central processing unit (CPU) implementation. Thus, our numerical results demonstrate the promising performance boosts that FPGAs are able to achieve over CPUs. As such, our proposed methods have the potential to serve as a building block for solving more complicated high-dimensional optimal control problems in real-time.

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arXiv2023

Lax-Oleinik-type formulas and efficient algorithms for certain high-dimensional optimal control problems

Two of the main challenges in optimal control are solving problems with state-dependent running costs and developing efficient numerical solvers that are computationally tractable in high dimension. In this paper, we provide analytical solutions to certain optimal control problems whose running cost depends on the state variable and with constraints on the control. We also provide Lax-Oleinik-type representation formulas for the corresponding Hamilton-Jacobi partial differential equations with state-dependent Hamiltonians. Additionally, we present an efficient, grid-free numerical solver based on our representation formulas, which is shown to scale linearly with the state dimension, and thus, to overcome the curse of dimensionality. Using existing optimization methods and the min-plus technique, we extend our numerical solvers to address more general classes of convex and nonconvex initial costs. We demonstrate the capabilities of our numerical solvers using implementations on a central processing unit (CPU) and a field-programmable gate array (FPGA). In several cases, our FPGA implementation obtains over a 10 times speedup compared to the CPU, which demonstrates the promising performance boosts FPGAs can achieve. Our numerical results show that our solvers have the potential to serve as a building block for solving broader classes of high-dimensional optimal control problems in real-time.

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arXiv2023

Leveraging Multi-time Hamilton-Jacobi PDEs for Certain Scientific Machine Learning Problems

Hamilton-Jacobi partial differential equations (HJ PDEs) have deep connections with a wide range of fields, including optimal control, differential games, and imaging sciences. By considering the time variable to be a higher dimensional quantity, HJ PDEs can be extended to the multi-time case. In this paper, we establish a novel theoretical connection between specific optimization problems arising in machine learning and the multi-time Hopf formula, which corresponds to a representation of the solution to certain multi-time HJ PDEs. Through this connection, we increase the interpretability of the training process of certain machine learning applications by showing that when we solve these learning problems, we also solve a multi-time HJ PDE and, by extension, its corresponding optimal control problem. As a first exploration of this connection, we develop the relation between the regularized linear regression problem and the Linear Quadratic Regulator (LQR). We then leverage our theoretical connection to adapt standard LQR solvers (namely, those based on the Riccati ordinary differential equations) to design new training approaches for machine learning. Finally, we provide some numerical examples that demonstrate the versatility and possible computational advantages of our Riccati-based approach in the context of continual learning, post-training calibration, transfer learning, and sparse dynamics identification.

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