Data-Driven Equation Discovery

Dynamic Material Characterization

Experimental set up for the non-destructive laser vibrometry tests

Figure: Experimental set up for the non-destructive laser vibrometry tests. The specimen (pink) is attached to a transducer (black housing) which produces a shear wave excitation that travels through the specimen. The laser doppler vibrometer (blue equipment) measures the specimenÌı

A 3-pane image showing successful recovery of a PDE.

Figure: Spatio-temporal data (a) measured from a laser vibrometry test and (b) simulated using a PDE discovered via WSINDy with the experimental data. The discovered PDE matches the experimental data closely and the difference between the two data sets is shown in (c).

Governing equations in the form of ordinary and partial differential equations are valuable models for physical systems. However they can be difficult to derive, making them unknown, particularly for complex systems. Our work in this area focuses on discovering ODEs and PDEs from data and leveraging the interpretability of these models to gain insights about systems of interest. In general, we approach this goal with the class of sparse identification of nonlinear dynamics (SINDy) methods which use sparse regression to identify relevant terms from a basis of candidate functions. Several advancements have made these methods more robust to noise, including denoising and using a weak formulation, and therefore more successful in discovering equations from noisy experimental data. An ongoing set of work combines the weak form of the SINDy for PDEs method (doi.org/10.1016/j.jcp.2021.110525) with data from non-destructive laser vibrometry experiments to discover equations and material properties for composite materials.Ìı

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Neural Network Based Discovery

A 3-pane image, where the first image shows a true solution, the second image shows noisy training data, and the 3rd image shows accurate recovery of the true solution.

For noisy data, partial differential equation (PDE) discovery is increasingly difficult, as the computation of numerical derivatives from this data is particularly sensitive to perturbations. Our work in this area uses a neural network-based approach to tackle this challenge. We represent the system state using one neural network similar to a Physics Informed Neural Network (PINN). As opposed to interpretable methods such as Weak SINDy, we model the underlying PDE with another neural network. We formulate the PDE discovery goal as a constrained optimization problem, where the objective function encourages fitting the observed data, while the constraints ensure the neural network satisfies the target PDE at a set of collocation points. These collocation points rely upon automatic differentiation of the neural network representation of the state. As opposed to other penalty-based approaches, we use a trust-region interior-point method to solve the resulting constrained optimization problem, promoting better adherence to the PDE constraints. We also investigate using classical numerical methods (such as finite differences) to solve these neural network PDEs, which is the effort of current work.

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Physics-informed Machine Learning

A 3-pane image showing the simulation of crack propagation
In the past few years. using neural networks to represent solutions to partial differential equations (PDE's) has become a topic of much interest in the computational mechanics community. Owing to the ease with which neural networks represent a wide range of functional forms, these methods are especially appealing for physical models which give rise to sharp gradients in the solution field. Whereas traditional PDE approximations (such as the finite element method) require either a large number of degrees of freedom or some a priori knowledge to capture localized solutions, neural networks have more flexible approximation properties. To use these methods, it is necessary to enforce boundary conditions and formulate the solution as an optimization problem. A variety of both penalty and constraint formulations exist for handling the boundary conditions, and there are also a variety of ways to phrase the optimization problem. We explore the pros and cons of the different problem statements through examples in heat transfer, elasticity and fracture mechanics, with aim of developing methods that are competitive in head-to-head comparisons with incumbent numerical techniques like the finite element method.