Learning Nonlinear Transient Dynamics in Fluid Flow Networks with the Neural Basis Method

Fluid flow networks are ubiquitous across engineered and natural systems, including pipeline infrastructure and biological vasculature. In many practical regimes, they are governed by nonlinear edgewise hyperbolic laws coupled through continuity and conservation at junctions. Learning their dynamics is important in many-query scenarios requiring rapid evaluation across unseen parametric settings. Conventional methods re-solve the system for each new query, resulting in prohibitive cost. Existing physics-informed neural approaches typically impose governing equations through penalty losses over generic neural approximators, rather than embedding the physical laws in their neural formulation. Such losses act only as optimization surrogates during training, with no guarantee that their reduction reduces solution error. Moreover, the resulting optimization problem is prone to ill-conditioning, particularly for hyperbolic systems. Building on the “representation-then-projection” principle of the neural basis method, we introduce a new approach for modeling fluid flow networks. Fixed neural bases define the representation space on each edge, while projection of the discretized equations determines the expansion coefficients, embedding edgewise laws into a conservative formulation. Edge subdomains are coupled through non-overlapping domain-decomposition. A neural operator then learns, directly in this reduced space, the parametric map to the corresponding expansion coefficients. On two representative network topologies, looped and branching, we reproduce references with relative $L^2$ errors below $0.02\%$, whereas the physics-informed neural network baseline fails. Across unseen boundary conditions in both interpolation and extrapolation, we maintain mean pressure errors below $1\%$ and mean flux errors below $3.3\%$, while the physics-informed DeepONet baseline reaches $8$–$18\%$ pressure errors and flux errors of $85$–$95\%$, failing to capture the underlying dynamics. The results demonstrate the accuracy and efficiency of the proposed framework for learning nonlinear transient dynamics in fluid flow networks, highlighting its potential for real-time monitoring, design exploration, and uncertainty quantification.