Physics-informed neural networks with hard and soft boundary conditions for problems in fluid dynamics

Licentiatavhandling, 2025

This thesis investigates the use of Physics-Informed Neural Networks (PINNs) for solving forward problems in fluid dynamics through two classical cases: linear waves and lid-driven cavity problem. The linear wave case, based on potential flow theory, involves periodic waves over a flat bed and serves to examine how PINNs handle periodic boundary conditions and smooth solutions. The lid-driven cavity problem, a benchmark for viscous incompressible flow, introduces sharp gradients and corner singularities, challenging the network’s ability to handle discontinuous boundary conditions. Together, these cases evaluate PINNs across both smooth and complex flow regimes.

For the linear wave problem, both hard and soft strategies were used to enforce periodic boundary conditions (PBCs). The vanilla PINN predicted the velocity field with 2.31 % error. A periodic input achieved a 0.16 % velocity error, while soft constraints performed slightly better (0.10 %) but did not fully enforce periodicity. Hard enforcement of the kinematic bottom boundary condition (KBBC) using trial functions satisfied the constraint at machine precision, though at the cost of increased velocity error (2.36 %). The network also predicted the angular frequency omega with 0.03 % error and maintained velocity errors below 0.2 % without labeled data.

In the cavity flow case, a baseline PINN with soft boundary enforcement performed well across most of the domain but struggled near the top corners due to singularities. To improve accuracy, two strategies were tested: trial functions for hard boundary conditions and spatially varying loss weights. Hard Constraints 1, which left the top boundary soft, led to a marginal improvement, reducing MAE(v) by about 8 %, while Hard Constraints 2, which enforced the top boundary with a trial function, worsened the results. Weighting strategies were more effective: applying omega_mw reduced MAE(u) by 63 % and MAE(v) by 68 %, and using both omega_F and omega_mw yielded the best accuracy, with reductions of 81 % in MAE(u) and 78 % in MAE(v). These findings highlight the challenge of handling singularities.

Overall, the study provides practical insights and quantitative benchmarks to guide the design and training of PINNs for fluid dynamics problems.