Elucidating the Conformal Structure of the Brinkman Penalisation Method for Geometry-Adapted, Structure-Preserving Operator Learning of Hamiltonian PDEs
Authors: Teo Deveney, Baige Xu, Takaharu Yaguchi
Organizations: Department of Computer Science, University of Bath, United Kingdom · Center for Advanced Intelligence Project, RIKEN, Japan · Kyushu University / Center for Advanced Intelligence Project, RIKEN, Japan
The Brinkman penalisation method embeds boundary-value problems on complex domains into a simple computational box by modeling the solid region as a strongly dissipative medium, avoiding body-fitted mesh generation. We show that multi-symplectic Hamiltonian PDEs regularised by Brinkman-type penalisation retain a multi-conformal symplectic structure under a compatibility condition linking the symplectic matrix and the penalisation projection. This yields an exact local conservation law, under which the multi-symplectic two-form is conserved in the fluid region and decays exponentially inside the solid. The linear wave equation with Brinkman friction and Maxwell's equations with artificial Ohmic conductivity satisfy this condition, with explicit modified Hamiltonian densities. Building on this, we propose (i) structure-preserving numerical integrators via Strang splitting that satisfy a discrete conformal conservation law, and (ii) conformal symplectic neural operators that interleave exact dissipative flows with learnable multi-symplectic evolution operators, allowing geometry-dependent operator learning. Numerical experiments on wave and electromagnetic scattering demonstrate that our methods reproduce correct local energy budgets and avoid unphysical energy drift, providing a principled framework for physics-consistent scientific machine learning on complex domains.
Figures & tables
Figure 1: One-dimensional wave equation with Brinkman penalisation at T=5.0 for η=0.1 , 0.02 , 0.005 (solid region shaded). Smaller η yields stronger reflection; η=0.005 approximates a rigid wall almost perfectly.
Figure 2: 2D acoustic wave scattering by a circular obstacle ( η=0.02 ) computed with the conformal multi-symplectic split scheme ( 26 ): snapshots at t=0 , 1.5 , 3.0 , 4.0 . The pulse is reflected by the obstacle while the field inside the solid decays on the fast time scale η .
Figure 3: Discrete energy H(t) for the 2D scattering problem of Figure 2 : exact budget H(0)−∫(χ/η)ut2 (black), the proposed conformal multi-symplectic splitting, RK4, and explicit Euler on the same grid and time step. The proposed scheme tracks the exact budget whereas explicit Euler exhibits catastrophic energy growth (leaving the frame).
Figure 4: Operator learning for the penalised wave equation with an airfoil-shaped scatterer. Top: ground truth, CSNO (proposed), and FNO predictions of the wave field. Bottom: discrete energy H versus time. The FNO energy explodes unnaturally, while the proposed conformal symplectic neural operator tracks the ground-truth energy closely.
Figure 5: TM Maxwell scattering of an Ez pulse by a circular conductor modeled by the conductivity penalisation ( σ=χ/η , η=0.02 ), computed with the conformal split-Yee scheme. By Remark 3.7 the configuration is equivalent to Figure 2 , and the scattering pattern coincides.
Figure 6: Fully three-dimensional Maxwell scattering by a conducting sphere via the conductivity penalisation ( η=0.02 , periodic Yee lattice): mid-plane slices of Ez . The field is expelled from the conductor on the time scale η .
Figure 7: Discrete electromagnetic energy for the 3D conducting-sphere problem: exact Poynting budget H(0)−∫(χ/η)∣E∣2 (black), the proposed conformal split-Yee scheme, RK4, and explicit Euler on the same lattice and time step. The proposed scheme tracks the budget while explicit Euler exhibits catastrophic energy growth (leaving the frame).
Figure 8: Operator learning for the conductivity-penalised 3D Maxwell equations with randomly sampled conducting obstacles. Top: mid-plane Ez at the final frame of a rollout on a held-out geometry. Left to right these show ground truth, electromagnetic CSNO (proposed), and FNO3d baseline. Bottom: electromagnetic energy H(t) along the rollout. The FNO energy decays spuriously to less than half of the true value, while the CSNO tracks the ground-truth energy to within 0.5% at the final time.
Learning solution operators for differential equations is a central problem in scientific machine learning. However, many neural operator methods optimize prediction accuracy without explicitly enforcing the geometric structure of the dynamics. Structure-preserving models such as SympNets and Symplectic Neural Flows address this issue for conservative Hamiltonian systems by preserving the symplectic form. In dissipative Hamiltonian systems with conformal symplectic structure, however, the symplectic form evolves according to a conformal factor determined by the dissipation. We propose CoSynFlow, a conformal symplectic neural flow for learning continuous-time solution maps of dissipative Hamiltonian dynamics. CoSynFlow composes symplectic shear maps with explicit conformal scaling, preserving the conformal symplectic structure by construction. By conditioning it on a finite-dimensional Hamiltonian descriptor and the dissipation parameter, a single trained model predicts solution maps for unseen systems without retraining. CoSynFlow keeps the structure error at machine precision, attains the lowest long-horizon error, and admits physics-informed training.
Baige Xu, Takaharu Yaguchi
RIKEN AIP · Motooka Nishi-ku, Fukuoka, Japan · IMI, Kyushu University
The modeling and simulation of infinite-dimensional Hamiltonian systems are central problems in mathematical physics and engineering, however they pose significant computational and structural challenges for standard data-driven architectures. In this work, we introduce the Symplectic Neural Operator, a neural operator architecture designed to preserve the symplectic structure intrinsic to Hamiltonian PDEs. We provide a theoretical characterization of their symplecticity and establish a rigorous long-term stability result based on the combination of symplectic structure preservation and learning accuracy. Numerical experiments on canonical Hamiltonian PDEs corroborate this theoretical result and show that SNOs exhibit improved energy behavior compared with non-structure-preserving neural operators.
Yeang Makara, Yusuke Tanaka, Takashi Matsubara +1
Graduate School of Science Kobe University 1-1 Rokkodai-cho, Nada-ku, Kobe, Japan · NTT Communication Science Laboratories 2-4, Hikaridai, Seika-cho, Soraku-gun, Kyoto, Japan · Faculty of Information Science and Technology Hokkaido University Kita 14, Nishi 9, Kita-ku, Sapporo, Hokkaido, Japan +1
We present the first method to directly use a learned continuous Lagrangian to forecast the dynamics of systems governed by partial differential equations, exploiting the inherent conservative structure to achieve stable long-range predictions. We develop an optimization-based integrator that minimizes the squared Euler--Lagrange residual via a mesh-free near-symplectic construction on local space-time patches. Different from integrators for analytical models, integrators for learned models should decouple model error (phase error) from integration error (conservation error). By relying on optimization rather than time-stepping, we bypass the global coupling inherent to fixed discretizations, which slows time- and space-stepping and complicates learning. Our method scales linearly with domain size via Jacobi iteration, and places no structural requirements on the learned network, allowing it to be coupled with existing physics-guided machine learning (ML) methods. We validate our approach on a learned representation of a double pendulum, a one-dimensional wave equation, and a two-dimensional wave equation. Our method achieves error comparable to classical symplectic methods while generalizing to spatially varying dynamics and arbitrary boundary conditions without retraining.
Lyra Zhornyak, Eric Forgoston, M. Ani Hsieh
University of Pennsylvania · Montclair State University