Physics‐Informed Machine Learning Method for Large‐Scale Data Assimilation Problems

We develop a physics-informed machine learning approach for large-scale data assimilation and parameter estimation apply it estimating transmissivity hydraulic head in the two-dimensional steady-state subsurface flow model of Hanford Site given synthetic measurements said variables. In our approach, we extend conditional Karhunen-Loéve expansion (PICKLE) method to modeling with unknown flux (Neumann) varying (time-dependent Dirichlet) boundary conditions. demonstrate that PICKLE is comparable accuracy standard maximum posteriori (MAP) method, but significantly faster than MAP problems. Both methods use mesh discretize computational domain. MAP, parameters states are discretized on mesh; therefore, size problem directly depends size. PICKLE, used evaluate residuals governing equation, while approximated by truncated expansions number controlled smoothness state fields, not For considered example, cost increases near linearly (as N1.15) grid nodes N, much N3.28). also show once trained one set Dirichlet conditions (i.e., river stage), provides accurate estimates any value stage).