AI Swiftly Solves Complex Control Problems Previously Needing Intensive Computation

Scientists are tackling the challenge of efficiently solving nonsmooth optimal control problems for linear partial differential equations, a significant hurdle in fields ranging from engineering to finance. Yongcun Song of Nanyang Technological University, Singapore, Xiaoming Yuan and Tianyou Zeng from The University of Hong Kong, Hong Kong SAR, China, and Hangrui Yue of Nankai University, Tianjin, China, present a novel approach called iUzawa-Net, developed in collaboration between their respective institutions. This research introduces the first solver capable of providing real-time solutions for this class of problems by integrating an inexact Uzawa method with learnable components, effectively replacing traditional numerical techniques. The team demonstrate universal approximation properties and asymptotic optimality, validating the iUzawa-Net’s efficiency with both elliptic and parabolic optimal control problems, and establishing a versatile framework for future advances in optimisation-informed deep learning for PDE-constrained optimisation.

These problems, crucial for modelling complex systems, often involve finding the best control strategy to achieve a desired outcome described by a PDE. Optimal control models governed by PDEs are fundamental to understanding systems across numerous scientific and engineering disciplines, with typical applications including optimising designs in aerodynamics, refining precision control in chemical processes, and developing targeted drug delivery systems in medicine. The iUzawa-Net specifically addresses a critical bottleneck in these areas: the difficulty of solving problems where the desired outcome is not easily described by smooth mathematical functions. Nonsmoothness arises frequently in optimal control, often due to constraints or regularisations imposed on the control variables to ensure properties like boundedness or sparsity. This innovation synergises model-based optimisation algorithms with data-driven deep learning, inheriting the strengths of both approaches and offering a versatile framework for a wider range of PDE-constrained optimisation challenges. The iUzawa-Net represents a significant step towards overcoming limitations in existing numerical methods, particularly when dealing with ‘nonsmooth’ problems where standard optimisation techniques falter. The methodology begins by unrolling an inexact Uzawa method, a technique for solving saddle point problems, and replacing conventional preconditioners and PDE solvers with learnable neural networks. This allows the network to directly approximate the iterative process of the Uzawa method, potentially accelerating convergence and enabling real-time solutions. To implement iUzawa-Net, the research team constructed a deep learning architecture where each layer represents an operator within the Uzawa iteration. Specifically, neural networks were designed to approximate the solution operator, S, and its adjoint, S, which map between the control and state spaces. These learned operators effectively replace the computationally expensive process of discretizing and solving the underlying PDE at each iteration, incorporating specifically designed layers to handle the inherent nonsmoothness of the optimal control problem. Theoretical analysis demonstrates that iUzawa-Net can approximate the solution operator of a class of nonsmooth optimal control problems to arbitrary accuracy with just two layers under mild conditions, establishing it as a universal approximator for these problems. Crucially, the research establishes algorithm tracking for any layer number L*, meaning the network’s outputs can be interpreted as inexact iterations of the Uzawa method. This tracking capability, combined with an optimisation perspective, guarantees asymptotic ε-optimality, where for any specified error tolerance ε, a sufficiently deep network maintains a solution within an ε-neighbourhood of the true optimal pair. Further analysis reveals that, under specific regularity assumptions, an algorithm tracking and weight tying iUzawa-Net achieves asymptotic ε-optimality on bounded sets without requiring compactness, a significant advantage in infinite-dimensional spaces. This weight tying, achieved through shared layer parameters, substantially reduces model complexity while maintaining numerical accuracy. Validation of the network’s numerical efficiency was conducted using both nonsmooth elliptic and parabolic optimal control problems, demonstrating the network’s potential for efficient and accurate solutions. The research demonstrates universal approximation properties and establishes asymptotic ε-optimality for the iUzawa-Net, indicating its potential to approximate solutions with guaranteed accuracy. Computationally, the iUzawa-Net requires only a single offline training phase, enabling real-time solutions during inference via a single forward pass, circumventing the nested iterations inherent in traditional numerical methods. The network also facilitates data-driven preconditioner design within an end-to-end trainable framework, accelerating convergence and improving adaptability across diverse problem settings, and can learn the solution operator even without explicit knowledge of certain problem parameters, given sufficient training data. This isn’t merely about faster computation; it’s about opening up possibilities for real-time control and optimisation, envisioning digital twins that can adapt and respond instantaneously to changing conditions. Consider designing more efficient aircraft wings, optimising energy grids, or even personalising medical treatments based on dynamic physiological data. However, the current demonstration focuses on a specific class of problems, and scaling this approach to truly high-dimensional and complex scenarios remains a challenge. The reliance on training data also introduces the usual caveats of deep learning, as generalisation to unseen scenarios isn’t guaranteed. Future work will likely explore ways to reduce the data requirements and enhance the robustness of the iUzawa-Net, signalling a shift towards a hybrid paradigm where the precision of model-based methods is combined with the adaptability of data-driven techniques, potentially revolutionising how we tackle computationally intensive scientific and engineering problems.