Extended parametric physics-informed neural networks (ePPINNs) for efficient solution of heat transfer and fluid flow problems

: Physics-informed neural networks (PINNs) often suffer from computational inefficiency versus conventional solvers for parametric partial differential equations (PDEs). This work proposes an extended parametric PINNs (ePPINNs), which overcomes this limitation by innovatively incorporating parameters as an additional input dimension. This approach not only facilitates the solution of parametric PDEs that are intractable for traditional methods due to a lack of generalized boundary conditions but also achieves unprecedented computational efficiency. Specifically, a single trained ePPINNs model can resolve an entire parametric space, providing continuous solutions for any parameter within the trained range (e.g., Ra ∈ [10 0 ,10 6 ]). We validate our method on benchmark cases including the Burgers equation, 1D/2D heat conduction equations, and 2D natural convection. For instance, in natural convection problems across six orders of Rayleigh number (10 0 ≤ Ra ≤ 10 6 ), the proposed method requires only 8.72 hours on a single GPU, representing a speedup of over 100× compared to the 1.30 hours per parameter for the traditional LRBF (CPU) and 3.65 hours per parameter for the original PINNs (GPU). oncurrently, ePPINNs generate accurate flow and heat transfer predictions for any parameter, demonstrating exceptional generalization. This framework thus transcends the accuracy-efficiency trade-off of original PINNs and establishes a new paradigm for parametric systems in scientific computing.