PENG9570 Final Report: 03 Computational Methods

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Writing Project:: PENG9570 Course Project - Writing Project - VIBE Draft Index:: Drafts - PENG9570 Course Project - VIBE Outline Partner:: 04 - Numerical Methods and Schemes - Outline 1 Previous Draft:: 03 - Computational Methods - Draft 6 - HUMAN Role:: Final report draft adapted from Overleaf final modifications

3. Computational Methods

All numerical work was carried out in NeuroFEM_Annotated.ipynb, and NeuroFEM_Sweep.ipynb. The first Jupyter notebook is a derivative of the supplied NeuroFEM notebook with additional commentary added to illustrate the computational methods corresponding with the theoretical methodology. From this we test the baseline reference performance and the forcing function switch (Fig. \ref{fig:myfig1a}). The second notebook reproduces the full methodology for a parameter sweep comparing NeuroFEM, analytic, and standard solver performance between different readout magnitudes, mesh counts, and neurons per node.

The annotated notebooks are executed end-to-end on the verified Python 3.12 .venv environment. Runtime dependencies are NumPy, SciPy, Matplotlib, MeshPy/Triangle, and tqdm; no GPU or neuromorphic hardware is used.

3.1 Reference Run

The reference configuration was: max_V $=0.002$, n_bdry $=128$, $N_{\mathrm{interior}}=1218$, NPM $=16$, jacobi_precondition $=\text{True}$, ${\gamma }_{pow2}=-6$, ${\lambda }_d=8$, ${\lambda }_v=16$, $k_p=4$, $k_i=16$, ${\sigma }_v=0.00225$, $\Delta t=2^{-12}$, and $N_{\mathrm{steps}}=50000$. The cached output neurofem_results_nmesh-1218_npm-16_nstep-50000.npz was used for all reported metrics; a forced re-run reproduced the same values up to the stochasticity of sigma_v * np.random.randn(...) in run_model.

The mesh is built by build_disk_mesh(max_V, n_bdry, ...). The Triangle mesh generator \cite{shewchukTriangleEngineering1996} supplies the unstructured disk mesh; local $P_1$ element integrals assemble the reduced stiffness matrix A_sys and mass matrix M over interior degrees of freedom. Boundary values are imposed strongly by omitting boundary nodes from the unknown vector.

Two right-hand sides (forcing functions) are used. f1 is constant and has the analytic disk reference solution; f2 is an offset forcing used to test runtime switching. With jacobi_precondition $=\text{True}$, both A_sys and M are diagonally rescaled before either solver is run. This preconditioning reduces the condition-number burden of the algebraic system without changing the continuum PDE. The direct baseline is solve_system_sparse_direct, which wraps SciPy’s spsolve and gives the discrete FEM reference for each right-hand side.

The reported NeuroFEM network uses NPM $=16$, meaning 16 neurons per mesh node and $N_{\mathrm{neurons}}=19488$. The fixed parameters are ${\gamma }_{pow2}=-6$, ${\lambda }_d=8$, ${\lambda }_v=16$, $k_p=4$, $k_i=16$, ${\sigma }_v=0.00225$, $\Delta t=2^{-12}$, $\mu =\nu =0$, and n_timesteps $=50000$. The constructor samples the sparse readout matrix gamma, assembles GTAG, omega_f, thresholds, resets, and the two bias vectors bias_f1, bias_f2, then stores them in an .npz parameter file.

run_model integrates the LIF-PI equations for 50,000 forward-Euler steps. At step 25,000 the bias is switched from bias_f1 to bias_f2, while the recurrent weights remain fixed. This matches the article’s claim that, once a spiking FEM network is built for a PDE, new problem instances can be solved by varying biases alone. The constant-forcing solution is the time average of output over the last 10,000 pre-switch timesteps.