Machine Learning with Hard Constraints:Physics-Constrained Constitutive Models with Neural ODEs and Diffusion

Our current constitutive models of material behavior fall short of being able to describe the mechanics of soft tissues. This is because soft tissues like skin and rubber, unlike traditional engineering materials, exhibit extremely nonlinear mechanical behavior and usually undergo large deformations. Developing accurate constitutive models for such materials requires using flexible tools at the forefront of science, such as machine learning methods. However, our past experiences show that it is crucial to incorporate physical knowledge in models of physical phenomena. The past few years has witnessed the rise of physics-informed models where the goal is to impose governing physical laws by incorporating them in the loss function. However, we argue that such "soft" constraints are not enough. This "persuasion" method has no theoretical guarantees on the satisfaction of physics and result in overly complicated loss functions that make training of the models cumbersome.

We propose imposing the relevant physical laws as "hard" constraints. In this approach the physics of the problem are "baked in" into the structure of the model preventing it from ever violating them. We demonstrate the power of this paradigm on a number of constitutive models of soft tissue, including hyperelasticity, viscoelasticity and continuum damage models.

We also argue that new uncertainty quantification strategies have to be developed to address the rise in dimensionality and the inherent symmetries present in most machine learning models compared to traditional constitutive models. We demonstrate that diffusion models can be used to construct a generative framework for physics-constrained hyperelastic constitutive models.