Hamiltonian bridge: A physics-driven generative framework for targeted pattern control

TL;DR

This study shows how it can systematically build in physical priors into a generative framework for pattern control in non-equilibrium systems across multiple length and time scales.

Abstract

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.

Figures (8)

• Figure 1: Algorithmic framework for targeted pattern control.(A) Physics-driven framework for pattern generation. Top-left panel shows the general form of conserved and non-conserved PDEs considered in the present study. Here, $\varphi$ can be a scalar or vector valued field, $\mathbf{v}$ is a known vector field modeling the intrinsic pattern forming capacity and $\mathbf{u}$ is the control flux field, $R$ is a given reaction field, and $S$ is the control reaction field as described in \ref{['eq:pattern_formation_general_form']}. The pattern control objective is formulated as the steering of the pattern generated by the forward dynamics towards a target pattern as described in \ref{['eq:opt_ctrl_pattern_formation_terminal_constraints']}. In the Eulerian-Lagrangian control formulation, the Eulerian field $\phi(t, {\bf z})$ is first discretized via the Smoothed Particle Hydrodynamics (SPH) method (see SI section S4) to Lagrangian dofs, ${\bf x}_i$ and $\phi_i$, which when deployed on the Eulerian PDE, leads to a set of coupled non-linear equations for the Lagrangian dofs (see SI section S4). The Lagrangian dofs are then steered via ${\bf u}$ and ${\bf v}$ to obtain the target field via the optimal control objective. The Eulerian-Lagrangian optimal control formulation can be viewed as navigation on a complex landscape as shown on the right. (B) Schematic of the Lagrangian-Eulerian optimal control method for pattern design. At time $t'$, consider a Lagrangian set of coordinates as shown in bottom-left corner which gives a Eulerian pattern via smoothing kernels. Utilizing the Lagrangian degrees of freedom (dofs), the Eulerian fields evolve in time via the forward dynamical equations. At the terminal time, the error between the target pattern and evolved pattern is computed and backpropagated (via deterministic or stochastic adjoint method) both across scales and in time, to obtain a control at time $t$, which allows the Lagrangian coordinates (and the Eulerian pattern) to evolve in time (for details see SI Section S4).
• Figure 2: Targeted phase separation.(A) Controlling the phase separation of two species (blue and white colors in a square lattice chequered board pattern) governed by the Allen-Cahn Equation (see \ref{['drops_eqn']} and surrounding text). The uncontrolled Allen-Cahn equation is given by $\frac{\partial \phi}{\partial t} =R$, where $R$ is the reaction field given by $R=- \frac{\delta E }{\delta \varphi}$, where $E(\varphi)=\int_{\Omega} \left[ U(\varphi(\mathbf{z})) + \frac{\epsilon}{2} \left\| \nabla \varphi(\mathbf{z}) \right\|^2 \right] d^2 \mathbf{z}$ and $U(\varphi)=(1-\varphi^2)^2$, $\epsilon=10^{-3}$. The Eulerian pattern error and the target pattern 'H' (inset) is shown on the rightmost panel. (B) Controlling the phase separation of two species (blue and white colors from a randomly distributed initial condition) governed by the Cahn-Hilliard equation (see \ref{['drops_eqn']} and surrounding text) . The uncontrolled Cahn-Hilliard equation is given by $\frac{\partial \phi}{\partial t} + \nabla \cdot \left( m(\phi) \mathbf{v} \right)=0$, where $\phi$ is the phase, $m(\varphi)=m$ is the mobility, and $\mathbf{v}$ is the known vector field given by $\mathbf{v}=-\nabla \frac{\delta E }{\delta \varphi}$, where $E(\varphi)=\int_{\Omega} \left[ U(\varphi(\mathbf{z})) + \frac{\epsilon}{2} \left\| \nabla \varphi(\mathbf{z}) \right\|^2 \right] d^2 \mathbf{z}$ and $U(\varphi)=(1-\varphi^2)^2$, $\epsilon=10^{-3}$. The Eulerian pattern error and the target pattern 'H' (inset) is shown on the rightmost panel.
• Figure 3: Geometry of transport paths for pattern control. Pattern forming capacity implied by the physics changes the geometry of the transport paths underlying optimal pattern evolution (see (8)-(9) and surrounding text). The row corresponding to reaction control shows the optimal interpolant in pattern space for the case of a non-conserved order parameter, corresponding to the nominal Hamiltonian $H_0 \left(\varphi, \lambda \right) = - \int_\Omega \frac{\lambda^2}{2\gamma_{\operatorname{v}}} d^2 \mathbf{z}$, and with intrinsic pattern forming capacity encoded in the Hamiltonian $H_1 \left(\varphi, \lambda \right) = \int_\Omega \lambda R d^2 \mathbf{z}$. The row corresponding to flux control shows the ordinary differential equation underlying the transport paths for the case of a conserved order parameter, corresponding to the nominal Hamiltonian $H_0 \left(\varphi, \lambda \right) = - \int_\Omega \frac{1}{2\gamma_{\rm u}} \varphi \left\| \nabla \lambda \right\|^2 ~d^2 \mathbf{z}$, where $\varphi$ is the time-dependent pattern function and $\lambda$ is the corresponding conjugate momentum density. The top-left panel corresponds to the zero passive reaction field which leads to a linear interpolation of the patterns. The top-middle panel corresponds to a non-linear interpolation of the pattern when the reaction field is present (control of Allen-Cahn). The bottom-middle panel corresponds to the addition of a Hamiltonian for the intrinsic pattern forming capacity $H_1\left(\varphi, \lambda \right) = - \int_\Omega m(\varphi) \nabla \lambda \cdot \nabla U ~d^2 \mathbf{z}$ (control of Cahn-Hilliard). The rightmost column corresponds to the comparison in numerical experiments. In the rightmost column, we plot the mean target error between the optimal interpolant and target pattern as a function of time for the reaction control case (top-right panel) corresponding to the Allen-Cahn dynamics, and the curvature of the transport paths as a function of time for flux control corresponding to the Cahn-Hilliard dynamics.
• Figure 4: Targeted morphogenesis. (A) Control of a reaction-diffusion system. Controlling a reaction-diffusion system comprising three species, from a randomly generated reaction matrix and initial conditions (see SI section S7 for details). The uncontrolled reaction-diffusion system is given by (see (11) and surrounding text) $\frac{\partial \phi_i}{\partial t} = D \Delta \phi_i + f_i(\phi)$, where $\phi(t, {\bf z}) = \left(\phi_1(t, {\bf z}), \phi_2(t, {\bf z}), \phi_3(t, {\bf z}) \right)$ is the vector of concentration of species $i=1,2,3$ at time $t$ and ${\bf z} \in \Omega$, $f_i$ is the reaction field for species $i$ and $D$ is the diffusion constant. The rightmost panel shows that the evolution of the Lagrangian squared error (normalized w.r.t. initial error) relative to target pattern decays appproximately exponentially (inset shows the $H$-shaped target pattern with the vertical and horizontal bars to be formed by the three individual species, respectively). (B) Control of cell fate dynamics. The three panels are the representative intermediate patterns corresponding to the controlled cell fate decision making model (see (12) and surrounding text). The forward uncontrolled equations are given by $\frac{\partial \rho}{\partial t} = \nabla \cdot \left[ \rho \left(\nabla \left( \frac{\delta E}{\delta \rho} \right)\right) \right]$ and $\frac{\partial \phi}{\partial t} - \left[\nabla \left( \frac{\delta E}{\delta \rho} \right) \right] \cdot \nabla \phi = - \frac{\delta E}{\delta \varphi} + R$ , where $E(\rho, \varphi) = \int_{\Omega} \left( f(\rho) + g(\rho) \left\| \nabla \varphi \right\|^2 \right) d^2 \mathbf{z}$. The figure on the left is the initial pluripotent state, the figure in the middle is the snapshot of the dynamical system at time $T/2$, and the rightmost figure is the snapshot of the dynamical system at time $T$. The rightmost panel shows that the evolution of the Lagrangian pattern error (normalized w.r.t. initial error), relative to target pattern decays appproximately exponentially.
• Figure S5: Targeted phase separation.(A) Model A: Allen-Cahn equation. The Eulerian pattern error and the target pattern 'H' (inset) is shown on the rightmost panel (inset showing the error on a logarithmic scale indicating exponential decay). (B) Model B: Cahn-Hilliard equation. The Eulerian pattern error and the target pattern 'H' (inset) is shown on the rightmost panel (inset showing the error on a log-log scale indicating power law decay).