Gradient-based optimization of basis function methods for solving nonlinear time-dependent PDEs

Somayeh Jafari

doi:10.1016/j.camwa.2026.06.025

Gradient-based optimization of basis function methods for solving nonlinear time-dependent PDEs

2026-07-06

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DOI：10.1016/j.camwa.2026.06.025

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We introduce Basis Function Optimization (BFOpt), a gradientbased framework for solving time–dependent nonlinear partial differential equations (PDEs). Unlike black–box neural–network solvers, BFOpt is a mesh–based method that approximates the solution by a combination of spatial basis functions. For the 2D heat equation (linear), BFOpt achieves a mean squared error (MSE) below 10−8 , outperforming deep finite difference methods (DFDM). For the 2D convection–diffusion equation (nonlinear) with Reynolds numbers R=100 , 200, and 300, BFOpt obtains a mean absolute error below 0.008 on a regular 60 × 60 grid in under one minute of CPU time – a regime where many numerical and deep learning methods require additional stabilization. The algorithm automatically identifies time–dependent parameters by observing which coefficients change across time steps. On 5000 scattered points, BFOpt solves the same equation from t=0 to t=1 in 250 seconds of CPU time, whereas a discrete–time PINN (Crank–Nicolson) requires 2000 seconds and reaches a much larger error (above 10−2 ), highlighting the computational and accuracy advantages of BFOpt. The method is stable for all tested time steps (Δt ≤ 0.01), and a learning–theoretic uniform β-stability analysis is provided.

Gradient-based optimization of basis function methods for solving nonlinear time-dependent PDEs

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