Physics-Informed Neural Networks as Differentiable Surrogates for 4D-Variational Data Assimilation
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 tangentlinear and adjoint models that are costly to derive and maintain alongside evolving forward codes, and that often fail for the non-differentiable formulations common in biogeochemical systems. 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 all three operators simultaneously from a single computational graph. Applied to nitrate-phytoplankton-zooplankton (NPZ) dynamics as a controlled testbed, our approach achieves a 6.5-fold speedup over traditional numerical models in twin experiments while maintaining equivalent state estimation accuracy across observation sampling scenarios. Jacobian analysis reveals how the network reconciles embedded structural constraints with training data, highlighting that the PINN preserves ecologically meaningful relationships while adapting 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.