The training process of many deep networks explores the same low-dimensional manifold.

We develop information-geometric techniques to analyze the trajectories of the predictions of deep networks during training. By examining the underlying high-dimensional probabilistic models, we reveal that the training process explores an effectively low-dimensional manifold. Networks with a wide range of architectures, sizes, trained using different optimization methods, regularization techniques, data augmentation techniques, and weight initializations lie on the same manifold in the prediction space. We study the details of this manifold to find that networks with different architectures follow distinguishable trajectories, but other factors have a minimal influence; larger networks train along a similar manifold as that of smaller networks, just faster; and networks initialized at very different parts of the prediction space converge to the solution along a similar manifold.

Bifurcation instructed design of multistate machines.

We propose a design paradigm for multistate machines where transitions from one state to another are organized by bifurcations of multiple equilibria of the energy landscape describing the collective interactions of the machine components. This design paradigm is attractive since, near bifurcations, small variations in a few control parameters can result in large changes to the system's state providing an emergent lever mechanism. Further, the topological configuration of transitions between states near such bifurcations ensures robust operation, making the machine less sensitive to fabrication errors and noise. To design such machines, we develop and implement a new efficient algorithm that searches for interactions between the machine components that give rise to energy landscapes with these bifurcation structures. We demonstrate a proof of concept for this approach by designing magnetoelastic machines whose motions are primarily guided by their magnetic energy landscapes and show that by operating near bifurcations we can achieve multiple transition pathways between states. This proof of concept demonstration illustrates the power of this approach, which could be especially useful for soft robotics and at the microscale where typical macroscale designs are difficult to implement.

Discontinuous Metric Programming in Liquid Crystalline Elastomers.

Liquid crystalline elastomers (LCEs) are shape-changing materials that exhibit large deformations in response to applied stimuli. Local control of the orientation of LCEs spatially directs the deformation of these materials to realize a spontaneous shape change in response to stimuli. Prior approaches to shape programming in LCEs utilize patterning techniques that involve the detailed inscription of spatially varying nematic fields to produce sheets. These patterned sheets deform into elaborate geometries with complex Gaussian curvatures. Here, we present an alternative approach to realize shape-morphing in LCEs where spatial patterning of the crosslink density locally regulates the material deformation magnitude on either side of a prescribed interface curve. We also present a simple mathematical model describing the behavior of these materials. Further experiments coupled with the mathematical model demonstrate the control of the sign of Gaussian curvature, which is used in combination with heat transfer effects to design LCEs that self-clean as a result of temperature-dependent actuation properties.

Multivalued Inverse Design: Multiple Surface Geometries from One Flat Sheet.

Designing flat sheets that can be made to deform into three-dimensional shapes is an area of intense research with applications in micromachines, soft robotics, and medical implants. Thus far, such sheets were designed to adopt a single target shape. Here, we show that through anisotropic deformation applied inhomogeneously throughout a sheet, it is possible to design a single sheet that can deform into multiple surface geometries upon different actuations. The key to our approach is development of an analytical method for solving this multivalued inverse problem. Such sheets open the door to fabricating machines that can perform complex tasks through cyclic transitions between multiple shapes. As a proof of concept, we design a simple swimmer capable of moving through a fluid at low Reynolds numbers.

Curved Geometries from Planar Director Fields: Solving the Two-Dimensional Inverse Problem.

Thin nematic elastomers, composite hydrogels, and plant tissues are among many systems that display uniform anisotropic deformation upon external actuation. In these materials, the spatial orientation variation of a local director field induces intricate global shape changes. Despite extensive recent efforts, to date there is no general solution to the inverse design problem: How to design a director field that deforms exactly into a desired surface geometry upon actuation, or whether such a field exists. In this work, we phrase this inverse problem as a hyperbolic system of differential equations. We prove that the inverse problem is locally integrable, provide an algorithm for its integration, and derive bounds on global solutions. We classify the set of director fields that deform into a given surface, thus paving the way to finding optimized fields.