Data-Driven Equation Â鶹ÊÓƵy
Dynamic Material Characterization
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.Ìý
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Neural Network Based Â鶹ÊÓƵy
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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