Enhancing neural network extrapolation in thermo-fluid systems using steady-state solutions

Abstract

Time-dependent partial differential equations (PDEs) arise in many engineering systems, including thermo-fluid applications. Classical numerical simulations of such systems can become computationally expensive for long-time dynamics because they typically require sequential time integration with time steps constrained by stability, accuracy, or nonlinear solvers. Although scientific machine learning provides an alternative for approximating PDE solutions, standard neural network approximations often degrade when extrapolated beyond the training time interval. In this work, we propose a steady-state-informed neural network representation for dissipative PDE systems whose solutions relax toward a stationary equilibrium. The proposed ansatz decomposes the solution into a steady-state component and a transient correction modulated by a time-dependent decay profile. When the decay profile vanishes at long time and the transient correction remains bounded, the representation embeds convergence to the prescribed steady state directly into the architecture, rather than enforcing it through an additional penalty term. This allows the network to learn the transient dynamics while preserving the correct asymptotic behavior. We implement the approach within a physics-informed neural network (PINN) framework and train the resulting model using the SOAP optimizer. The method is evaluated on a sequence of problems of increasing physical and geometric complexity, ranging from the one-dimensional heat equation to incompressible Navier-Stokes flow in a lid-driven cavity, natural convection in a square cavity, and a full three-dimensional conjugate heat transfer problem. The numerical results show that the steady-state-informed architecture substantially improves temporal extrapolation beyond the training interval compared with architectures that do not explicitly enforce the asymptotic condition.