Fractional diffusion solvers stop paying for memory at every step: precomputed Hermite weights bring relative L2 error to 1.98×10⁻⁵
Synopsis
A neural collocation scheme for semilinear time-fractional reaction–diffusion equations rewrites the Caputo memory term as cubic-Hermite product integration, using the network state and its autodifferentiated time slope as endpoint data on each time panel while the weakly singular weights are precomputed once before training. On the scalar benchmark v(t)=t4.2 the Hermite memory approximation is three to six orders of magnitude more accurate than the classical L1 formula on the tested meshes, and for a manufactured problem with α=0.7 a small CPU-trained network reaches a relative L2 error of 1.98×10⁻⁵ and a maximum error of 3.80×10⁻⁵.
Interpretation
Neural solution of time-fractional reaction–diffusion equations moves the memory term out of the training loop: the history integral of the Caputo derivative becomes a weighted sum over precomputed panel weights, so each optimizer step only accumulates history. Earlier fractional physics-informed networks combined automatic differentiation with numerical fractional differentiation, yet the memory discretization remained the decisive accuracy bottleneck because automatic differentiation does not remove the history integral. On the scalar benchmark v(t)=t4.2 at t=1, across α of 0.4, 0.7 and 0.9 and meshes from N=8 to N=64, the cubic-Hermite absolute error is three to six orders of magnitude smaller than the L1 formula on the same mesh.
Initial and homogeneous Dirichlet data are built into the network output, leaving only the PDE residual in the training loss and removing penalty weights for those two constraints. Multi-term PINN objectives can be poorly balanced, and boundary penalty weights must be chosen by the user. On the manufactured problem with α=0.7, κ=0.05, λ=1 and T=1, a network with 25 interior points, 16 time intervals and three hidden layers of 32 tanh units, trained by 1500 Adam iterations plus L-BFGS, reaches a relative discrete L2 error of 1.98×10⁻⁵ and a maximum pointwise error of 3.80×10⁻⁵ on a 201×101 test grid, with a training-grid residual MSE of 2.01×10⁻⁸.
The same memory operator also serves the inverse problem, jointly estimating fractional order and diffusivity by minimizing the RMS discrete PDE residual. The inverse experiment uses a clean synthetic field from the manufactured state so that parameter identifiability is separated from network approximation error. From four separated initial guesses (0.30,0.12), (0.55,0.08), (0.85,0.02) and (0.90,0.15), every run converges to the true pair (0.7,0.05), with 10, 8, 7 and 10 function evaluations.
Perspective
The demonstrated scope is one-dimensional spatial domains and semilinear reaction–diffusion equations with a Caputo time derivative of order 0<α<1, on a uniform temporal grid, with a manufactured solution for the forward problem and a noise-free synthetic field for the inverse problem. The author notes that a graded temporal mesh can be inserted without changing the neural formulation, and lists multidimensional domains and noisy inverse problems as directions for extension.
The consistency estimate assumes smooth temporal traces, while fractional diffusion solutions can show an initial weak singularity even for smooth data, so the estimate describes the smooth-trace regime. The inverse experiment deliberately uses clean synthetic data to isolate identifiability, and the author makes no claim about robustness to substantial observation noise. For large Nt the history assembly remains O(Nt²) unless fast convolution or compression is added.