L. Mahadevan

Vishaal Krishnan, Sumit Sinha, L. Mahadevan

Patterns arise spontaneously in a range of systems spanning the sciences, and their study typically focuses on mechanisms to understand their evolution in space-time. Increasingly, there has been a transition towards controlling these patterns in various functional settings, with implications for engineering. Here, we combine our knowledge of a general class of dynamical laws for pattern formation in non-equilibrium systems, and the power of stochastic optimal control approaches to present a framework that allows us to control patterns at multiple scales, which we dub the "Hamiltonian bridge". We use a mapping between stochastic many-body Lagrangian physics and deterministic Eulerian pattern forming PDEs to leverage our recent approach utilizing the Feynman-Kac-based adjoint path integral formulation for the control of interacting particles and generalize this to the active control of patterning fields. We demonstrate the applicability of our computational framework via numerical experiments on the control of phase separation with and without a conserved order parameter, self-assembly of fluid droplets, coupled reaction-diffusion equations and finally a phenomenological model for spatio-temporal tissue differentiation. We interpret our numerical experiments in terms of a theoretical understanding of how the underlying physics shapes the geometry of the pattern manifold, altering the transport paths of patterns and the nature of pattern interpolation. We finally conclude by showing how optimal control can be utilized to generate complex patterns via an iterative control protocol over pattern forming pdes which can be casted as gradient flows. All together, our study shows how we can systematically build in physical priors into a generative framework for pattern control in non-equilibrium systems across multiple length and time scales.

F. Giardina, L. Mahadevan

Walking is a common bipedal and quadrupedal gait and is often associated with terrestrial and aquatic organisms. Inspired by recent evidence of the neural underpinnings of primitive aquatic walking in the little skate Leucoraja erinacea, we introduce a theoretical model of aquatic walking that reveals robust and efficient gaits with modest requirements for body morphology and control. The model predicts undulatory behavior of the system body with a regular foot placement pattern which is also observed in the animal, and additionally predicts the existence of gait bistability between two states, one with a large energetic cost for locomotion and another associated with almost no energetic cost. We show that these can be discovered using a simple reinforcement learning scheme. To test these theoretical frameworks, we built a bipedal robot and show that its behaviors are similar to those of our minimal model: its gait is also periodic and exhibits bistability, with a low efficiency gait separated from a high efficiency gait by a "jump" transition. Overall, our study highlights the physical constraints on the evolution of walking and provides a guide for the design of efficient biomimetic robots.

Guido Novati, Lakshminarayanan Mahadevan, Petros Koumoutsakos

Controlled gliding is one of the most energetically efficient modes of transportation for natural and human powered fliers. Here we demonstrate that gliding and landing strategies with different optimality criteria can be identified through deep reinforcement learning without explicit knowledge of the underlying physics. We combine a two dimensional model of a controlled elliptical body with deep reinforcement learning (D-RL) to achieve gliding with either minimum energy expenditure, or fastest time of arrival, at a predetermined location. In both cases the gliding trajectories are smooth, although energy/time optimal strategies are distinguished by small/high frequency actuations. We examine the effects of the ellipse's shape and weight on the optimal policies for controlled gliding. Surprisingly, we find that the model-free reinforcement learning leads to more robust gliding than model-based optimal control strategies with a modest additional computational cost. We also demonstrate that the gliders with D-RL can generalize their strategies to reach the target location from previously unseen starting positions. The model-free character and robustness of D-RL suggests a promising framework for developing mechanical devices capable of exploiting complex flow environments.