Comparison of Computational Methods for Solving Inverse Problems in Fluid Mechanics Using Traditional and Machine Learning Approaches

July 6, 2026

In recent years, advances in computational methods and machine learning have provided powerful frameworks for solving inverse problems in fluid mechanics and heat transfer, particularly where data is sparse and system parameters are unknown. This thesis conducts a systematic comparison between the widely adopted Physics-Informed Neural Networks (PINN) and the novel Optimising a Discrete Loss (ODIL) framework. While PINN enforce physical laws continuously via automatic differentiation, ODIL embeds these constraints through discrete numerical residuals on a grid. Both frameworks are applied to benchmark problems governed by hyperbolic~(Wave), elliptic~(Poisson), and parabolic~(unsteady Heat) equations, culminating in a complex 2D reaction-diffusion problem with a spatially dependent, unknown source term. Results demonstrate that while PINNs offer a flexible mesh-free approach, they frequently suffer from optimization stiffness and spectral bias in multi-scale scenarios. To mitigate this, Fourier feature mappings were introduced to the PINN architecture (STMsFFN, MsFFN), improving accuracy by up to two orders of magnitude, albeit at a higher computational cost dependent on the mapping complexity. Conversely, ODIL, particularly when enhanced with multigrid decomposition, consistently outperformed PINN in both computational efficiency and accuracy, accelerating convergence by one to two orders of magnitude in stiff problems. Furthermore, hybrid strategies integrating neural networks into the ODIL framework for parameter inference proved highly effective. This study concludes that ODIL provides a robust, stable, and significantly more efficient alternative to continuous physics-informed frameworks for solving ill-posed engineering problems.

Authors in the community:

• 

  V

  Vasco Lopes de Carvalho

  ist199566

Supervisors of this institution:

• 

  J

  José Carlos Fernandes Pereira

  ist11668

mechanical-engineering - Mechanical engineering

eng - English

Instituto Superior Técnico