Data-driven models such as those common in machine learning tend to be relatively domain-agnostic and thus are widely-applicable across many domains.  It is a fundamental challenge how best to combine such data-driven models with fine-scale domain-driven models that are common in physics and other natural sciences and that make strong use of domain-specific insight.  Among other things, a domain-informed model formulation should encode some degree of stability or robustness or well-conditioning (in that a small change of the input will not lead to drastic changes in the output), characteristic of the underlying scientific problem.  Here, we describe recent work on using techniques from dynamical systems theory to combine these two types of models in a principled way.  We’ll describe how to develop physics-informed autoencoders using Lyapunov stability, leading to novel domain-driven regularization and to models that improve the generalization error and reduce prediction uncertainty for fluid flow problems.  We’ll also describe ContinuousNet, a variant of the popular residual neural network (ResNet) model that is meaningfully continuous-in-depth.  By embedding discrete neural network models into higher-order numerical integration schemes, e.g., Runge Kutta schemes, ContinuousNet can learn to represent continuous dynamical systems (which ResNets and other nominally-continuous models cannot); and, by exploiting ideas from numerical integration theory, ContinuousNet have improved robustness properties as well as improved training and inference properties on standard (non-scientific) machine learning tasks.