Domain decomposition: XPINN, cPINN, FBPINN

A fixed-capacity MLP cannot resolve features at all scales simultaneously. As the domain grows or the solution gets more features, single-network PINNs underfit. Three decomposition families emerged to scale PINN to large or complex problems:

The three families

• cPINN (Jagtap, Kawaguchi, Karniadakis 2020). Split the domain into non-overlapping subdomains; one network per subdomain; conservative coupling along the interfaces enforces continuity of the solution and the flux. Best for hyperbolic conservation laws (Euler, shallow water).
• XPINN (Jagtap, Karniadakis 2020). Generalisation of cPINN to arbitrary subdomain decomposition with continuity-loss interface coupling (no flux constraint). Best for parabolic and elliptic problems where conservation is not the primary issue.
• FBPINN (Moseley, Markham, Nissen-Meyer 2023). Finite-Basis PINN. Use overlapping subdomains with a smooth partition-of-unity weighting $w_k(x)$ that satisfies $\sum_k w_k(x) = 1$ for all $x$. The global solution is $u(x) = \sum_k w_k(x) M_k(x)$. No interface loss is needed — the windowing makes the global solution smooth by construction. Best for problems with multiple localised features.

FBPINN is the most modern and the most elegant. The interface loss in cPINN/XPINN is a third loss term that introduces its own loss-balance crisis (§3.2–§3.3). FBPINN avoids this by encoding the smoothness in the architecture.

The FBPINN ansatz

For a 1D domain $[a, b]$ split into $K$ overlapping windows centred at $c_k$ with half-widths $h_k$, define the cosine-square window

and the partition-of-unity weights $w_k(x) = \rho_k(x) / \sum_j \rho_j(x)$. Each subnet $M_k(x)$ is a small MLP. The global solution is

For a PDE $u''(x) = f(x)$ we need $u''$: by the chain rule

The window derivatives $w_k'$ and $w_k''$ are computed analytically (we know $\rho_k$ in closed form); each subnet provides $M_k'$ and $M_k''$ via the same forward-mode AD machinery from Part 0. The PDE-residual gradient flows back into every subnet whose window covers $x$ — just three or four subnets at any point thanks to the windowing.

Try it: FBPINN vs single MLP

The widget solves $u''(x) = f(x)$ with five sharp Gaussian-bump source terms at $x = -0.7, -0.35, 0, 0.35, 0.7$. A single 1-32-32-1 MLP races a 6-window FBPINN where each subnet is a 1-16-16-1 MLP (FBPINN has more total parameters but the per-subnet capacity is small — each window covers ~one bump). Watch the per-window weights $w_k(x)$ in the right panel and the final $u(x)$ fit on the left.

What you should observe

• Single MLP (1-32-32-1, ~1.1k params): given enough epochs, can actually fit this 1D problem cleanly — relative-L² typically a fraction of a percent. The single MLP is a strong baseline on 1D problems.
• FBPINN (6 × 1-16-16-1, ~3k total params, but only ~500 active per point): each subnet specialises on its window; the partition-of-unity enforces a smooth global solution. Relative-L² typically a few percent — competitive but not dramatically better on this 1D toy.
• The window panel shows the six cosine-square partition functions: $w_1$ is large near $x = -1$, $w_6$ near $x = 1$, with smooth blending in between. They sum to 1 everywhere.

Important honest note. On this small 1D toy a 32-32 single MLP can already fit five sharp bumps cleanly. The FBPINN advantage emerges in 2D and 3D, where a single network of comparable size cannot resolve all features simultaneously. The widget shows the mechanism — partition-of-unity windowing, analytic window derivatives, per-subnet PDE-residual flow — in the smallest setting where it can be implemented. The pedagogical purpose is the construction; the empirical advantage is unlocked by scale.

Why FBPINN scales

The killer feature of FBPINN is parallelism. Each subnet trains independently except for the partition-of-unity coupling at boundaries. On a 100-window decomposition, each subnet can run on its own GPU (Moseley et al. 2023 demonstrate up to 1024-window decompositions on multi-GPU clusters). The decomposition is the only PINN technique that genuinely scales to billion-parameter PDE problems — which is roughly the cost of a 3D acoustic FWI on a Sleipner-class survey.

Choosing windows

• Number $K$: enough that each window resolves a single dominant feature. For a multi-bump 1D problem, $K$ = number of bumps + a few extras.
• Half-width $h$: large enough that adjacent windows overlap on at least one collocation point. Smaller $h$ means more specialisation per subnet but more total subnets.
• Subnet capacity: small networks are fine; the windowing is the architectural lift, not the subnet width.

Expertise checkpoint — end of Part 3

You should now be able to:

• Diagnose a stalled PINN training by reading the per-term loss trace, identifying the named pathology, and reaching for the correct remedy.
• Implement loss-weight tuning, NTK-balanced weighting, gradient-norm balancing, SA-PINN, causality weighting, frequency continuation, RAR/RAD, and FBPINN — at least on toy problems.
• Decide which remedies a real seismic-PINN problem needs (typically NTK + causality + RAR + curriculum together).
• Defend why FBPINN is the recommended decomposition for large 2D and 3D wave equations.
• Critique training-failure complaints in any current PINN paper using the language of this Part.

Part 4 sets up the wave equations in PINN form. Many of the tools from this Part will be visible in every demo there.

References

• Jagtap, A.D., Kawaguchi, K., Karniadakis, G.E. (2020). Conservative physics-informed neural networks on discrete domains for conservation laws. CMAME 365, 113028.
• Jagtap, A.D., Karniadakis, G.E. (2020). Extended physics-informed neural networks (XPINNs). CICP 28(5), 2002–2041.
• Moseley, B., Markham, A., Nissen-Meyer, T. (2023). Finite basis physics-informed neural networks (FBPINNs). Adv. Comput. Math. 49, 62.
• Dolean, V., Heinlein, A., Mishra, S., Moseley, B. (2024). Multi-level FBPINNs for high-frequency multi-scale problems. JCP.