From Micro-physics to Stable Macro-models: Variational Learning for Non-Newtonian Fluids

Machine-learned PDE models provide a promising approach to predict in-sample dynamics, yet their numerical robustness and out-of-distribution generalization remain open challenges, especially when physical structure is not enforced. We introduce a general approach for learning stable, interpretable macroscale PDEs by constructing the energy variational structure directly from microscale physical laws. In this framework, we introduce a set of micro-macro encoders to model the unresolved micro-physics as generalized field variables, along with an extendable energy functional and variational form that strictly preserve the conservation laws and entropy production. We illustrate this approach through the non-Newtonian hydrodynamics of polymeric fluids, a canonical multiscale problem where conventional empirical closures often fail. The resulting model naturally inherits microscale structural-dependent, nonlinear interactions that challenge empirical closures. More importantly, the variational informed formulation guarantees frame-indifference objectivity, free energy decay, and positivity-preserving dynamics; various pre-existing energy-stable numerical schemes can be used to establish long-time simulations. In contrast, the direct PDE form-based learning leads to models that may fit training data but fail beyond it due to instability and loss of physical fidelity.