Scientific Machine Learning (SciML) is reshaping how we model, optimize, and control complex engineering systems by combining physics-based models with differentiable programming and modern machine learning. This talk presents a unified perspective on learning-to-control (L2C) and learning-to-optimize (L2O) as two complementary paradigms for constructing fast, scalable, and reliable real-time decision-making policies.

The L2C part focuses on Differentiable Predictive Control (DPC), a self-supervised policy optimization algorithm inspired by model predictive control that enables end-to-end training of parametric feedback policies via differentiable simulators. We present recent advances in learning controllers for families of optimal control problems governed by partial differential equations (PDEs). By integrating DPC with differentiable PDE solvers and neural operators, policies are learned directly as mappings between function spaces, enabling cardinality-invariant multi-agent control and zero-shot generalization across varying spatial discretizations, sensor layouts, actuator configurations, and agent populations.

This formulation addresses the scalability challenges inherent to infinite-dimensional PDE control by learning control operators in function spaces rather than discretization-specific policies.

The second part focuses on L2O for large-scale mixed-integer nonlinear programming (MINLPs) problems. We present a self-supervised learning algorithm that combines differentiable integer rounding with feasibility-restoring projection operators, allowing learning a high-quality approximate parametric solution policies while maintaining feasibility guarantees. The resulting policies achieve up to five orders of magnitude faster inference than conventional optimization while preserving solution quality across broad families of parametric MINLPs.

Together, these advances illustrate how differentiable programming enables a unified SciML framework for constructing scalable solution operators for continuous and discrete decision-making problems, opening new opportunities for real-time optimization and control of complex cyber-physical systems.

References:

[1] J. Drgoňa, A. Tuor and D. Vrabie, “Learning Constrained Parametric Differentiable Predictive Control Policies With Guarantees,” in IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 54, no. 6, pp. 3596-3607, June 2024

[2] Pietro Zanotta, Dibakar Roy Sarkar, Honghui Zheng, Somdatta Goswami, Ján Drgoňa, Cardinality-Invariant Neural Operator Policies for Scalable PDE Control, ICML, 2026

[3] Bo Tang, Elias B. Khalil, Ján Drgoňa, Learning to Optimize for Mixed-Integer Non-linear Programming with Feasibility Guarantees, arXiv:2410.11061, 2025, under review

[4] Ján Boldocký, Shahriar Dadras Javan, Martin Gulan, Martin Mönnigmann, Ján Drgoňa, Learning to solve parametric mixed-integer optimal control problems via differentiable predictive control, arXiv:2506.19646, 2025, to appear at the IFAC World Congress

[5] SciML tutorial at L4DC 2026 https://sciml-l4dc2026.github.io/SciML-L4DC2026/