Coupling Feel++ & SciMBA for Advanced PDE Solvers

[lyyric]

Coupling Feel++ & SciMBA for Advanced PDE Solvers

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Project 1 – Reserved

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Project 2 – Reserved

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Project 3 – Hybrid FEM + PINN Domain Decomposition

Goal
Couple a classical Finite Element solver (Feel++) with a Physics-Informed Neural Network (SciMBA) on complementary subdomains. A non-overlapping Schwarz iteration enforces continuity of temperature and heat flux at the interface.

Problem Description

One-dimensional heat equation on the spatial interval $[0,1]$ and time interval $[0,0.03]$:

$\partial_t u - \alpha,\partial_{xx}u = 0,\quad u(x,0)=\sin(\pi x),\quad u(0,t)=u(1,t)=0.$

Domain Partition

Subinterval	Method	Notes
$[0,0.4]$	FEM (Feel++)	Backward Euler, $h = 0.005$
$[0.4,1]$	PINN (SciMBA)	SIREN network, 4000 pts/iter

Iteration Scheme

Non‐overlapping Schwarz (Dirichlet ↔ Neumann) with relaxation factor $\omega = 0.8$.

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Key Result

After just 3 Schwarz iterations, the jump at the interface falls below $10^{-3}$; iterations 4–5 hit the FEM truncation/PINN approximation error floor.

Figure – exact solution; hybrid solver result after 5 iterations. The vertical gray line marks the interface at x = 0.4 .

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Error Metrics

Iteration $k$	Interface $L^2$ Error	Global $L^2$	Global $H^1$
1	$3.23\times10^{-1}$	$8.70\times10^{-1}$	56.8
2	$1.12\times10^{-2}$	$1.74\times10^{-1}$	11.4
3	$7.78\times10^{-2}$	$4.50\times10^{-2}$	2.32
4	$9.12\times10^{-2}$	$3.10\times10^{-2}$	0.59
5	$9.39\times10^{-2}$	$3.00\times10^{-2}$	0.36

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Take-aways

• The interface jump drops from $\mathcal{O}(10^{-1})$ in iteration 1 to $\mathcal{O}(10^{-2})$ in iteration 2, and falls below $\mathcal{O}(10^{-3})$ by iteration 3.
• By iteration 3, the concatenated solution is effectively $C^0$-continuous at $x = 0.4$; any remaining slope mismatch is on the order of $10^{-3}$ – $10^{-2}$.
• Iterations 4–5 yield no significant improvement because FEM truncation error and PINN approximation error both lie around $10^{-3}$ – $10^{-2}$, reaching a numerical “floor.”
• To reduce the interface error further, one must refine the FEM mesh (reduce $\delta_{\mathrm{FEM}}$) or improve PINN accuracy (reduce $\delta_{\mathrm{PINN}}$), for example by using automatic differentiation for $\partial_x u$ or enlarging the PINN network.

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Relevant Links

• Feel++ (FEM library) – https://github.com/feelpp/feelpp
• SciMBA (Scientific ML library) – https://sciml.gitlabpages.inria.fr/scimba
• Coupling Prototype Repository – https://github.com/feelpp/feelpp-scimba

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