Physics-Informed Neural Networks as Differentiable Surrogates for 4D-Variational Data Assimilation

Abstract

Biogeochemical forecasting requires state estimation methods that accommodate sparse observations, nonlinear dynamics, and discontinuous ecological processes. Four-dimensional variational data assimilation (4D-Var) addresses this challenge but depends on tangent-linear and adjoint models that are costly to derive and maintain alongside evolving forward codes, and that are not defined where biogeochemical formulations are non-differentiable. We propose a framework in which a physics-informed neural network (PINN) replaces the traditional numerical model and its linearizations, remaining differentiable through automatic differentiation while embedding simplified governing equations as structural guardrails and computing the forward, tangent-linear, and adjoint operators simultaneously from a single computational graph. Applied to nitrate-phytoplankton-zooplankton (NPZ) dynamics as a controlled testbed, our approach achieves an approximately six-fold speedup over traditional numerical models in twin experiments while maintaining comparable state estimation accuracy across observation sampling scenarios. Jacobian analysis shows that the learned sensitivities are consistent with the embedded ecological relationships where these agree with the data, while departing where simplified formulations prove inadequate. This framework demonstrates that PINN-based surrogates can maintain dynamical consistency in variational data assimilation while reducing implementation burden, offering a path toward more flexible and maintainable forecasting systems for increasingly complex biogeochemical models.