Finite Basis PINNs

DLSC Project B, with Guglielmo Pacifico — a study and partial reproduction of Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations.

The problem

Standard PINNs struggle with high-frequency and multi-scale problems — a manifestation of the spectral bias of neural networks. As the target $u(x)$ contains more frequencies, the network has to grow rapidly, and the optimisation landscape gets correspondingly worse.

What FBPINNs do

The domain $\Omega$ is split into $n$ partially overlapping subdomains. Each subdomain gets its own small network $NN_i$, and the global solution is a sum of windowed local solutions:

$$\overline{NN}(x;\theta) = \sum_{i=1}^{n} w_i(x) \cdot \mathrm{unnorm} \circ NN_i \circ \mathrm{norm}_i(x)$$

where $w_i(x)$ is a smooth window built from sigmoid factors, and $\mathrm{norm}_i$ rescales each subdomain to $[-1,1]$. The boundary condition is baked into an ansatz $\hat{u}(x;\theta) = \tanh(\omega_n x)\, NN(x;\theta)$ so the loss can collapse to the physics term alone:

$$\mathcal{L}(\theta) = \frac{1}{N_p} \sum_{i=1}^{N_p} \big\| \mathcal{D}\hat{u}(x_i;\theta) - f(x_i) \big\|^2$$

Experiments

We reproduced the headline experiments from the paper:

• Single-frequency case — $du/dx = \cos(\omega x)$ for $\omega = 1$ and $\omega = 15$. PINNs match FBPINNs at low frequency; at $\omega = 15$ the PINN underfits while the FBPINN tracks the analytical $\frac{1}{\omega}\sin(\omega x)$.
• Multi-scale case — $du/dx = \omega_1 \cos(\omega_1 x) + \omega_2 \cos(\omega_2 x)$ with $\omega_1=1, \omega_2=15$. Five subdomain networks of $2 \times 16$ neurons recover the full solution; the equivalent PINN needs $5 \times 128$ to get close.
• Scalability — varied the number of frequency components $n \in \{2,3,5,8\}$ and tracked $L_1$-loss vs training time. PINN training time grows roughly linearly with depth; FBPINN time grows weakly with subdomain count once you parallelise.

The image on the card is the per-subdomain output $w_i(x) \cdot NN_i(x)$ — overlapping localised wave packets that sum to a smooth global solution.

Takeaways

• For mildly multi-scale problems ($n \approx 3$) a deep PINN gets there — the FBPINN's win is mostly time, not accuracy.
• For genuinely multi-scale problems the subdomain decomposition seems to break the spectral bias by giving each network a band-limited target.
• The window function setup matters more than expected. We saw qualitatively different individual-network solutions vs the paper, even with consistent global behaviour, suggesting the decomposition is somewhat under-determined.