Domain decomposition

Overview

A single physics-informed network uses one global function for the whole domain. For a material with sharp phase transitions, the largest errors appear at those transitions, and the gradients near them can become hard to optimise 2110.07374 §3 2110.07374 §5.1. Domain decomposition answers this by cutting the domain into subdomains and giving each one its own network. The networks are coupled by interface terms that require neighbouring subdomains to agree on displacement and traction across their shared edge 2110.07374 §5.2.

The invariant the variants share is that the local equations hold inside each subdomain, and the solution is continuous across the interfaces. The paper calls its version a conservative PINN (cPINN), in the sense of Jagtap, Kharazmi and Karniadakis (2020), which has no card here 2110.07374 §5.2.

Variants

Whole-domain PINN (no decomposition)

One network covers the entire domain. It is the baseline against which the split versions are compared. In the paper's convergence study it reaches a mean residual of 1.49 x 10^-1 in the split comparison, against 1.22 x 10^-1 for the 4x4 cPINN 2110.07374 fig. 10. Card: Physics informed neural networks for continuum micromechanics.

Conservative PINN (cPINN) on a regular grid

The domain is split into a regular grid of subdomains, each with its own fully connected network. Each subdomain keeps the local residual loss. The interface terms add, for each shared edge, squared differences of the displacement components and of the stress components that carry the traction across that edge 2110.07374 §5.2. The paper uses a 4x4 split, with weight ψ = 20 on the interface terms 2110.07374 §5.3. Card: Physics informed neural networks for continuum micromechanics.

The paper tests grids from 1x1 to 5x5 and reports that the 5x5 split has a higher mean residual than the 4x4 split. Its explanation is that each extra subnetwork adds interface terms that make the loss harder to optimise 2110.07374 fig. 10 2110.07374 §5.3.

Subdomain networks with adapted boundary constraints

Subnetworks that share part of the outer boundary need hard boundary constraints adapted to that case, and the paper states this as a requirement 2110.07374 §5.2. The same paper applies its 4x4 cPINN to a micro-CT slice of a composite, with 90 units per subnetwork [arxiv:2110.07374 §6.2.2]. The release does not include the CT data, so this case cannot be rerun from the code 2110.07374 §6.1. All of the paper's results are author-reported.

When to use which

The paper's evidence is one problem family: 2D linear elastostatics on a square with a stiff inclusion, plus a micro-CT slice 2110.07374 §4.3 2110.07374 §6. Before using the split count or ψ = 20 elsewhere, check them on your own problem. The paper does not claim they transfer.

Implementation sketch

Language-agnostic outline of the cPINN loss. Each subdomain has its own network and a local residual loss; each shared edge adds interface terms once. The paper leaves open whether a shared edge should be counted once or twice, and the released code counts it once 2110.07374 §5.2. See the card's code notes.

# Build a regular split of the square into n_x by n_y subdomains  [§5.3]
# Each subdomain gets its own network  [§5.2]
def build_subdomains(n_x, n_y):
    subdomains = []
    for i in range(n_x):
        for j in range(n_y):
            # Four hidden layers  [§4.3.1]; width is a free choice  [§5.3]
            net = fully_connected(depth=4, width=64 // n_x)
            subdomains.append((net, bounds(i, j)))
    return subdomains

# Total cPINN loss: local residuals plus weighted interface agreement  [§5.2]
# psi = 20 in the paper's experiments  [§5.3]
def cpinn_loss(subdomains, interfaces, psi=20.0):
    total = 0.0
    for net, box in subdomains:
        # Local PDE residual loss inside this subdomain  [§4.1]
        total = total + local_residual_loss(net, box)
    for edge in interfaces:
        # One term per shared edge, counted once  [§5.2]
        a, b = edge.left, edge.right
        if edge.is_horizontal:
            # Displacement u_x and the traction stresses must agree  [§5.2]
            total = total + psi * mse(a.u_x(edge.pts), b.u_x(edge.pts))
            total = total + psi * mse(a.sigma_xx(edge.pts), b.sigma_xx(edge.pts))
            total = total + psi * mse(a.sigma_xy(edge.pts), b.sigma_xy(edge.pts))
        else:
            # Vertical edge: u_y and the other traction components  [§5.2]
            total = total + psi * mse(a.u_y(edge.pts), b.u_y(edge.pts))
            total = total + psi * mse(a.sigma_yy(edge.pts), b.sigma_yy(edge.pts))
            total = total + psi * mse(a.sigma_xy(edge.pts), b.sigma_xy(edge.pts))
    return total

Notes on where variants differ. The paper's loss is minimised with full BFGS rather than Adam 2110.07374 §4.1. Clipping the gradient is used by the adaptive scheme in the text, and the threshold is not stated 2110.07374 §5.1. The paper says only that the subnetwork width was chosen to keep about 13 x 10^3 weights in total 2110.07374 §5.3. The sketch's width of 64 // n_x is the released code's rule, not a rule the paper states, see the card's code notes.

Contradictions

• Size of the cPINN gain. The text says the 4x4 cPINN reduces the mean residual by one quarter relative to the PINN 2110.07374 §5.3. The plotted values give 1.22 x 10^-1 against 1.49 x 10^-1, about 18% lower 2110.07374 fig. 10. The card's pitfalls advise using the figure values.
• Adaptive PINN versus PINN. The text says the adaptive PINN is lower than the PINN in the first half of the study. The table shows this only at 16^2 and 32^2 points; at 8^2 the adaptive PINN is higher 2110.07374 fig. 8. This is about adaptive collocation, but it affects the comparison in the split study.
• Parameter count for the CT slice. The paper gives about 2.7 x 10^4 weights for the CT cPINN 2110.07374 §6.2.2. The card's own count from the layer sizes is about 2.5 x 10^4 per subnetwork and about 4 x 10^5 for 16 subnetworks. The paper does not say which count it means, and the card's pitfalls note the gap.

Papers

• Physics informed neural networks for continuum micromechanics: the cPINN with 4x4 subdomains, weighted interface agreement, and the split and adaptive comparisons. Released code is GPL-3.0 and differs from the paper in several settings.