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.

翻译：模式在跨越多个科学领域的系统中自发产生，对其研究通常聚焦于理解其时-空演化的机制。近年来，逐渐出现了在各种功能场景中控制这些模式的趋势，这对工程应用具有重要意义。本文结合了我们对非平衡系统中模式形成的一般动力学规律的认识，以及随机最优控制方法的能力，提出了一种能够在多尺度上控制模式的框架，我们称之为"哈密顿桥"。我们利用随机多体拉格朗日物理与确定性欧拉模式形成偏微分方程之间的映射关系，借鉴我们最近提出的基于Feynman-Kac伴随路径积分公式的相互作用粒子控制方法，并将其推广至图案化场的主动控制。我们通过数值实验展示了该计算框架的适用性，包括含与不含守恒序参量的相分离控制、流体液滴的自组装、耦合反应-扩散方程以及时空组织分化的唯象模型。我们从理论角度阐释数值实验结果，理解底层物理如何塑造模式流形的几何结构，从而改变模式的传输路径与模式插值的性质。最后，我们通过展示如何利用最优控制，在可表述为梯度流的模式形成偏微分方程上实施迭代控制协议以生成复杂模式，作为本文的总结。综合而言，我们的研究表明，如何系统地将物理先验知识融入生成框架，以实现跨越多重时空尺度的非平衡系统模式控制。