Full State Estimation from Sparse Measurements

In order to use Partial Differential Equations (PDEs) for forecasting, one must start with an initial state. In most practical applications, the full state of the system is not known, and instead only a small number of observations is available. The problem then becomes, how can we use these measurements to estimate the full state of the system?

To achieve this, we leveraged a PDE, more specifically the nonlinear Schrödinger equation, to obtain synthetic data to train a recurrent neural network (RNN) to predict the optimal coefficients associated with modes from the proper orthogonal decomposition (POD). Using this pretrained RNN, we then used actual, experimental data, from this study, to not only estimate the full state of the system, but also to make forecasts using NLS. We were able to successfully predict the formation of a 2-soliton solution, which is thought to be connected to the formation of rogue waves on the oceans surface.

Z. T. Hilliard, M. Farazmand, and A. Chabchoub. Full state estimation of deep-water waves from sparse measurements: Predicting rogue waves from experimental data. (In preparation, Exp. 2025)

Sequential Data Assimilation for Shape-Morphing Solutions (DA-SMS)

PDEs are an integral tool for making forecasts. However various forms of errors can limit the predictive power of the associated model. Data assimilation seeks to mitigate these issues by incorporating observational data into the numerical approximation of the PDE. However, many existing methods are very computationally expensive. To address both of these issues, we use a non-linear approximation of the solution to the PDE that incorporates observational data into the approximation.

We have successfully applied DA-SMS to a few problems including the nonlinear Schrödinger equation, the Kuramoto-Sivashinsky equation (depicted below), and the 2D advection-diffusion equation.

Sequential data assimilation for PDEs using shape-morphing solutions, Z. T. Hilliard, M. Farazmand. J. Comput. Phys. vol. 533, pp. 113994, 2025.

Enforcing Conserved Quantities in Galerkin Truncation and Finite Volume Discretization

Many PDEs have associated conserved quantities such as total mass, energy, enstrophy etc. that the solution must satisfy. However, it is well known that these physical properties of the solution may not survive numerical approximation. We developed methodologies known as Galerkin Reduced-Order Nonlinear Solutions (G-RONS) and finite-volume reduced-order nonlinear solutions (FV-RONS) which ensures that these associated conserved quantities are respected by the numerical solution to the PDE.

We successfully applied this methodology to the Nonlinear Schrödinger equation and the shallow water equation. Our methodologies and results can be found in this paper:

Enforcing conserved quantities in Galerkin truncation and finite volume discretization, Z. T. Hilliard, M. Farazmand, Nonlinear Dynamics, vol. 112, pp. 14051-14069, 2024