Program

Introduction to Physics-Informed Machine Learning:
Challenges and Opportunities

Thomas Beckers (Vanderbilt University)

Integrating Physics in Learning for Optimization-Based Control: From Models to Closed-Loop Guarantees

Rolf Findeisen, Maik Pfefferkorn, Sebastian Gasche, Hendrik Alsmeier (TU Darmstadt)

Optimization-based control, and in particular model predictive control (MPC), provides a flexible framework to handle constraints, optimize performance, and account for system dynamics. Its success, however, hinges on the availability of accurate and structured models of the system, its environment, and the control objectives. This talk explores how physics-informed machine learning can support and enhance MPC by embedding physical knowledge, structure, and constraints into learning-based models such as Gaussian processes and neural networks. We show how such models can capture not only system dynamics, but also disturbances, reference trajectories, and cost functions in a data-efficient and physically consistent manner, and how they can be tightly integrated within the MPC framework while preserving stability, safety, and constraint satisfaction. Moving beyond model learning, we then consider learning in the closed loop: safely adapting controller parameters, cost functions, and policies from operational data using stability-informed Bayesian optimization and related techniques. We conclude with an outlook on physics-informed generative models, for prediction and control. Throughout, examples from robotics, autonomous systems, and energy systems illustrate that combining physical structure with data-driven adaptation yields controllers that are both flexible and certifiable.

How Can We Leverage Physical Priors for Learning and Control?

Thomas Beckers (Vanderbilt University) 

Learning-based methods offer powerful tools for modeling and control of complex dynamical systems, but purely data-driven approaches can be data-intensive and may fail to respect known physical principles. This talk explores how physical prior knowledge can be integrated with machine learning to obtain more reliable and trustworthy learning-based control.

I will discuss four complementary approaches: discrepancy bias, where learning captures unknown residual dynamics; inductive bias, where physical structure is embedded directly into the model; observational bias, where physics is incorporated through physically consistent training data; and learning bias, where physical knowledge guides the training process through the loss function. Through examples from learning and control of dynamical systems, I illustrate how these approaches can combine the flexibility of machine learning with the structure and guarantees provided by physics.

Learning Lagrangian Models for Control in Unconventional Robots

Cosimo Della Santina (TU Delft)

Robots are increasingly moving beyond rigid, fully actuated mechanisms toward systems with compliant, deformable, and underactuated bodies. These unconventional designs offer new capabilities but also challenge classical model-based control: their dynamics are strongly nonlinear, difficult to derive accurately, and often too complex to use directly in feedback. In this talk, I will discuss how learning can help recover control-oriented models while preserving mechanical system structure.

The central idea is to learn Lagrangian models rather than arbitrary state-transition maps. By embedding properties such as inertia, potential energy, dissipation, and actuation structure into the learning architecture, we obtain models that remain interpretable and compatible with established nonlinear control tools.

Beyond established Lagrangian Neural Networks, I will focus on a new class of models, called coupled oscillator networks, which learn Lagrangian dynamics directly from pixels. This provides a step toward physics-consistent world models that combine perception, prediction, and control within a common mechanical representation.

Fast and Safe Model Predictive Control with Neural Approximations and Learned Iterative Solvers

Sergio Lucia  (TU Dortmund)

Model predictive control has become a powerful framework for decision-making in dynamic systems, but its computational demands still limit its widespread deployment, especially for nonlinear systems subject to uncertainty. One promising direction is to use neural networks to approximate optimal control laws and enable real-time implementation. However, this approach raises two central challenges: the large amount of data typically required for training, and the fact that the safety, robustness, and stability properties of the original MPC formulation are not automatically inherited by the resulting closed-loop system.

In this talk, I will discuss recent developments addressing these challenges. First, I will show how parametric sensitivities can be incorporated into the training process to significantly improve data efficiency and obtain accurate neural approximations with fewer samples. Second, I will present probabilistic and deterministic verification techniques that allow us to certify relevant closed-loop properties despite approximation errors and additional uncertainty. Finally, I will discuss recent advances in learning iterative steps of constrained optimization solvers, which aim to combine the speed of learning-based methods with the accuracy and reliability of optimization-based control.

The proposed techniques will be illustrated through simulation and experimental case studies involving both linear and nonlinear systems.

Learning Surrogate Models for Optimization and Control

Alberto Bemporad (IMT School for Advanced Studies Lucca)

In this talk, I will review several recently developed learning-based methods for optimization and control. First, I will present an approach for approximating solution mappings for broad classes of multiparametric nonlinear optimization and generalized Nash equilibrium problems. Next, I will introduce an active-learning method for nonlinear minimax regression that provides worst-case error guarantees for surrogate models of objective functions, feasible sets, and control-oriented dynamical systems. Finally, I will describe a framework for approximating the objective and/or constraint functions of nonconvex multiparametric programs as the minimum of a finite collection of convex functions, enabling the computation of approximate global solutions by solving only a finite number of independent convex optimization problems.

MPC as a Physics-Informed Policy: Reinforcement Learning over Model Predictive Control

Dirk Reinhardt (Norwegian University of Science and Technology)

Model Predictive Control (MPC) is, at its core, a physics-informed policy: it embeds a dynamics model, physical constraints, and an economic objective into a constrained optimization problem that is solved online at every time step. This makes it a natural meeting point for physics and data. In this talk, we show how viewing MPC through the lens of Markov Decision Processes (MDPs) turns it from a fixed control law into a structured, differentiable policy class whose every component — the prediction model, the cost function, the constraints, and the terminal conditions — can be learned from closed-loop data with reinforcement learning (RL).

We focus on a question that is as practical as it is theoretical: why does learning *over* an MPC controller work? The key is a paradigm shift away from fitting the most accurate dynamics model and toward directly optimizing closed-loop performance, while the physics-informed structure of the MPC keeps the resulting policy interpretable, constraint-satisfying, and data-efficient. Under this holistic parametrization, a possibly inexact physical model acts as a strong prior that data refines rather than replaces.

The talk is accompanied by a companion coding exercise built on "leap-c", an open-source Python framework that couples the fast differentiable MPC solver "acados" with "PyTorch". Working through a building climate-control task, participants construct and train a learnable MPC controller and observe how data-driven tuning improves closed-loop energy performance while respecting comfort and physical constraints.

Learning to Control PDEs with Differentiable Predictive Control and Neural Operators

Ján Drgoňa (Johns Hopkins University)

Scientific machine learning is creating new opportunities for controlling systems governed by partial differential equations (PDEs). In this talk, we present a differentiable programming methodology that combines Differentiable Predictive Control (DPC) with neural operators for end-to-end learning of feedback policies for PDE-constrained control. DPC reformulates parametric model predictive control as an offline gradient-based policy optimization problem, enabling policies to be optimized by backpropagating task objectives and constraint penalties through differentiable PDE solvers and surrogates. To overcome the limitations of fixed-dimensional policy representations, we further cast PDE control as an operator learning problem that maps state fields to continuous control functions, enabling policies that naturally adapt to varying sensor, actuator, and multi-agent configurations. Remarkably, policies trained on small agent populations exhibit cardinality invariance, enabling zero-shot transfer to significantly larger populations and robustness to partial agent failure. We demonstrate the methodology on tracking, stabilization, and density transport across linear, nonlinear, chaotic, and turbulent PDE systems.