Shu-Xia Tang
August 24, 2025- San Diego, CA, USA
Learning, Estimation & Control for PDE Systems
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About The Workshop
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This full-day workshop explores recent advances at the intersection of machine learning, estimation, and control for systems governed by partial differential equations (PDEs). The event brings together experts from academia and industry to address scalable and theory-grounded methods for PDE-based modeling, observer design, and real-time decision-making. The program features two thematic sessions: Learning, Control, and Decision with PDEs, and Battery Modeling and Estimation with PDEs. Topics include neural operator methods, boundary control, data-driven estimation, and multiphysics applications in mobility and energy infrastructure. The workshop provides a collaborative forum to exchange ideas and foster interdisciplinary research at the frontiers of control theory, applied mathematics, and cyber-physical systems.​
Organizers
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Schedule of The Workshop
Morning Session
Learning, Control, and Decision with PDEs
9:00- 10:00 AM
Supervised vs. unsupervised Learning for First Order Hyperbolic Nonlinear PDEs: application to traffic flow modeling
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Alexandre Bayen, University of California, Berkeley.
This talk investigates the use of neural networks (NNs) as discrete approximations of first order nonlinear hyperbolic partial differential equations (PDEs). Two approaches are considered. The first is a supervised approach, performed on solutions of a training set composed of Riemann problems. The second approach, unsupervised, follows the physically inspired neural network (PINN) approach, in which a loss function based on the initial PDE is used to train the neural network. When available, semi-explicit solutions are used for validation (i.e. Lax-Hopf formula), which enables the comparison of these two approaches against classical numerical schemes (i.e. Lax-Friedrichs, Godunov, ENO, WENO, discrete Galerkin, etc.). The need of the Kruzhkov entropy condition is discussed in the context of unsupervised learning. Efficiency of methods (accuracy, computational cost) are discussed. Finally, the supervised approach is demonstrated on field data collected by a drone (Berkeley Deep Drive dataset); predictive capabilities of the supervised approach is assessed with success in comparison to all other discretizations of the underlying PDEs. Applications of the methods are finally presented in the context of mixed-autonomy traffic in which self-driving vehicles are used to control the entirety of traffic.
Keynote Presentation Information:
Regular Presentation Information:
10:30- 11:00 AM
Machine Learning in Service of PDE Backstepping Control
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​​​​​ Mamadou Diagne, University of California, San Diego.
This talk presents recent advances in leveraging Machine Learning to enhance the practical implementation of PDE backstepping control design. Specifically, we employ deep neural networks, Deep Operator Networks (DeepONets), to accelerate the computation of gain kernel functions in the control law. DeepONets, which approximate nonlinear operator mappings (function-to-function), have demonstrated the ability to encode complete PDE control strategies, including backstepping. Consequently, for any new functional coefficient in a PDE system, the associated backstepping gains can be efficiently retrieved via a single function evaluation. I will introduce a framework for approximating multiple coupled nonlinear operators that emerge in the backstepping boundary control of counter-convective hyperbolic PDE systems. Such systems arise in diverse domains, including oil drilling dynamics, the Saint-Venant model for shallow water waves, the Aw-Rascle-Zhang model for traffic flow, and gas pipeline transport. Assuming fullstate measurements, the proposed method offers theoretical guarantees, notably global exponential stability in the L2 sense, even when employing approximate gain kernels. The analysis highlights that inaccurate kernel approximation—e.g., due to limited training data—can adversely affect performance by increasing overshoot and reducing the system’s decay rate.
11:00- 11:30 AM
Neural Operator Learning for Nonlinear Delay Systems with Predictor Feedback
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​ Yuanyuan Shi, University of California, San Diego.
Predictor feedback designs are critical for delay-compensating controllers in nonlinear systems. For stability analysis, input delays can be modeled as a transport PDE, and the closedloop delay system with predictor feedback can be analyzed as a coupled ODE-PDE system. However, predictor feedback designs are limited in practical applications, as numerical solutions of the predictor are computationally expensive for nonlinear systems. In this talk, I will present our recent advance on approximating the predictor mapping via neural operator (NeuralOP) learning. To analyze the proposed NeuralOP predictor feedback design in closed-loop systems, we present a new analytical framework via PDE backstepping transformation. We show that under the NeuralOP predictor feedback, the resulting delay system achieves semiglobal practical stability (dependent on the approximation error). We conduct experiments controlling a 5-link robotic manipulator with different state-of-the-art NeuralOP architectures, demonstrating speedups on the magnitude of 102 compared to traditional predictor approximation schemes.
11:30 AM- 12:00 PM
A Model-Based Approach for Continuous-Time Policy Evaluation with Unknown
L´evy Process Dynamics
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​​ Xiaochuan Tian, University of California, San Diego.
Reinforcement learning (RL) is active branch of machine learning focused on learning optimal policies to maximize cumulative rewards through interaction with the environment. While traditional RL research primarily deals with Markov decision processes in discrete time and space, we explore RL in a continuous-time framework, essential for high-frequency interactions such as stock trading and autonomous driving. Our research introduces a PDE-based framework for policy evaluation in continuous-time environments, where dynamics are modeled by L´evy processes. We also formulate the Hamilton-Jacobi-Bellman (HJB) equation for the corresponding stochastic optimal control problems governed by L´evy dynamics. Our approach includes two primary components: 1) Estimating parameters of L´evy processes from observed data, and 2) Evaluating policies by solving the associated integro-PDEs. In the first step, we use a fast solver for the fractional Fokker-Planck equation to accurately approximate transition probabilities. We demonstrate that combining this method with importance sampling techniques is vital for parameter recovery in heavy-tailed data distributions. In the second step, we offer a theoretical guarantee on the accuracy of policy evaluation considering modeling error. Our work establishes a foundation for continuous-time RL in environments characterized by complex, heavy-tailed dynamics.
12:00 - 12:30 PM
Feedback Control of Nonlinear Systems Using The Polynomial-Polynomial Regulator
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​​​ Nicholas A. Corbin, University of California, San Diego.
In this work, we present recent advancements in tensor-based algorithms for computing Taylor approximations to the optimal control law for high-dimensional nonlinear systems. The approach is built on the theory of Al’brekht; however, emphasis is placed on overcoming longstanding scalability limitations which have historically restricted Al’brekht’s method to lowdimensional examples. The result is a controller which we call the polynomial-polynomial regulator (PPR), a generalization of the linear-quadratic regulator (LQR) to polynomial dynamics with polynomial cost functions. We provide a scalable and open-source Matlab implementation of our function ppr(), a nonlinear analog to Matlab’s lqr() that returns the polynomial coefficients of the feedback law. These polynomial control laws 1) more accurately approximate the optimal control law than linear approximations, and 2) are able to stabilize some nonlinear systems that fail to be stabilized by linear controllers. PPR controllers are demonstrated on a variety of examples including semidiscretized PDEs leading to high-dimensional nonlinear systems that illustrate the scalability of the proposed approach.
Afternoon Session
Battery Modeling and Estimation with PDEs
1:30 PM- 2:30 PM
Keynote Presentation Information:
This talk highlights advances in the control and estimation of battery systems through partial differential equation (PDE) modeling and control theory. Utilizing the Single Particle Model with Electrolyte (SPMe), a class of quasilinear parabolic PDEs, the talk describes estimation frameworks for state-of-charge (SOC) and thermal state diagnostics. Key contributions include the design of Lyapunov-based adaptive observers for PDE systems, ensuring asymptotic convergence of state and fault estimates under parametric uncertainty. Stability analyses leverage backstepping transformations and Lyapunov functions to provide theoretical guarantees for the accuracy of the estimators. Experimental results validate the proposed methods, achieving temperature estimation errors within 0.2°C and fault detection times under five seconds. These advancements enable batteries to operate at their physical limits safely, with implications for fast-charging and reliability in industrial applications. This work demonstrates the potential of PDE-based control theory to enhance the safety, efficiency, and longevity of energy storage technologies.
Regular Presentation Information:
3:00 PM- 3:30 PM
Cascaded Electrochemical-Thermal Modeling and Temperature Estimation for Cylindrical Li-ion Batteries
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​ Shu-Xia Tang, Texas Tech University.
Cascaded electrochemical-thermal modeling is essential for understanding the intricate temperature dynamics of lithium-ion batteries, which directly impact their safety, performance, and longevity. This study introduces a comprehensive multi-layer model that captures detailed thermal and electrochemical interactions across all layers and components of a cylindrical lithium-ion battery. While the multi-layer model provides precise temperature predictions and might identify potential hotspots, its complexity makes real-time applications challenging. To address this, a simplified single-layer model is developed, focusing on a single representative layer with eight components. This model reduces computational demands, enabling efficient real-time temperature estimation. Experimental validation demonstrates the superior accuracy of the multi-layer mode compared to the single-layer model, which trades detail for simplicity. To bridge the gap between accuracy and practicality, an optimized sensor placement strategy is proposed, supported by Luenberger observer to estimate battery case temperatures using experimental data. These findings underscore the potential of the proposed models to enhance thermal safety, optimize monitoring, and guide the development of next-generation battery technologies.
3:30 PM- 4:00 PM
Resilient Thermal Management of Batteries Using Control Barrier Functions
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​ Tanushree Roy, Texas Tech University.
The vision of smart cities has been gaining attention worldwide as human civilization grows increasingly urban-centric. To meet the demands of optimal performance in such cities, reliable operation of various energy systems is crucial. However, with increased automation, connectivity, and complexity in these energy systems, they are becoming more vulnerable to anomalies such as faults and cyberattacks. This talk will be focused on the resilient operation of battery energy storage systems using algorithms designed from control-theoretic tools to guarantee an appropriate level of fidelity under faults and cyberattacks. Specifically, I will discuss the design of anomaly-resilient thermal management using control barrier function approach for lithium-ion batteries.
4:00 PM- 4:30 PM
Balancing Accuracy, Observability, and Identifiability in Electrochemical Battery Models​
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​ Joseph N. E. Lucero, Stanford University.
Developing accurate and computationally efficient electrochemical models is essential for advancing battery management systems. Typically formulated as partial differential equations (PDEs), these electrochemical models present fundamental challenges in balancing numerical accuracy, nonlinear observability, and parameter identifiability. This presentation examines how model parameterization, numerical spatial discretization, as well as output equation assumptions impact state estimation accuracy and the model’s structural identifiability. We highlight tradeoffs in the spatial discretization of PDE-based battery models, demonstrating that finer resolution grids do not necessarily enhance nonlinear observability or improve the estimation quality of model-based observers. Furthermore, we discuss the challenges of nonlinear system identification for these models, particularly the difficulty of uniquely determining model parameters from experimental cycling data and the risks associated with parameter compensation, which may lead to misleading parameter values and inaccurate predictions of internal battery states. By addressing these fundamental issues, this work provides insights into how electrochemical models can achieve both predictive accuracy and practical applicability, ensuring reliable battery state estimation in real-world applications.
4:30 PM- 5:00 PM
Boundary observers for coupled diffusion–reaction systems​
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Leobardo Camacho Solorio, Apple, Inc.
We address the problem of state estimation for coupled linear diffusion-reaction partial differential equations (PDEs) with Neumann boundary conditions. Our approach involves the design of an observer that guarantees a prescribed convergence rate. Using a pair of integral transformations, we map the estimation error system to a target system with provable stability. This framework also extends naturally to the dual problem of boundary stabilization for the same class of PDEs. For numerical implementation, we introduce a scheme based on power series approximations of the transformation kernels, carefully accounting for their piecewise differentiability.