Cova-PINN: Cross-Domain Conservation Physics-Informed Neural Network for Fluid-Solid Conjugate Heat Transfer in Complex Geometries

Weizheng Zhang; Xunjie Xie; Hao Pan; Lin Lu

cs.LGOct 8, 2026

Cova-PINN: Cross-Domain Conservation Physics-Informed Neural Network for Fluid-Solid Conjugate Heat Transfer in Complex Geometries

Authors: Weizheng Zhang, Xunjie Xie, Hao Pan, Lin Lu

Organizations: Shandong University · Tsinghua University

Abstract

Multi-domain physics-informed neural networks (PINNs) flexibly model medium-specific representations to solve fluid--solid conjugate heat transfer (CHT). However, standard multi-domain PINNs enforce governing equations and interface conditions on separately sampled domain supports, which can yield plausible temperature fields but inaccurate end-to-end energy transfer and outlet temperatures. We propose Cova-PINN, a multi-domain PINN framework that aligns conservation support with thermal interaction paths in complex geometries. Cova-PINN jointly optimizes cross-domain composite control-volume balances at the local scale and paired-wall closure at the global exchanger scale. We evaluate Cova-PINN on four triply periodic minimal surface (TPMS) heat exchangers and a geometrically distinct DualMS design against CHT-specific, optimization-oriented, and complex-geometry PINN baselines under a common protocol. Relative to the closest baseline, MUSA-PINN-CHT, Cova-PINN reduces average outlet-temperature and device-level closure errors across the four TPMS topologies by $37.7\%$ and $60.2\%$ , respectively, while also improving full-field and heat-duty accuracy, with consistent gains on DualMS.

Figures & tables

Figure 1: Overview of Cova-PINN. (a) Energy follows the hot-fluid–solid–cold-fluid path, while separately supported pointwise residuals need not close its device-scale balance. (b) After independently training and freezing the two MUSA-PINN flow networks, the three temperature networks are jointly optimized by pointwise CHT physics and coordinated local–global conservation objectives, with annealed solid-mean regularization during early training.

		Primitive (P)			Gyroid (G)			Diamond (D)			IWP
Method	Ref.	$E_{\mathrm{field}}$ (K)	$E_{\mathrm{out}}$ (K)	$E_{\mathrm{clo}}$ (%)	$E_{\mathrm{field}}$ (K)	$E_{\mathrm{out}}$ (K)	$E_{\mathrm{clo}}$ (%)	$E_{\mathrm{field}}$ (K)	$E_{\mathrm{out}}$ (K)	$E_{\mathrm{clo}}$ (%)	$E_{\mathrm{field}}$ (K)	$E_{\mathrm{out}}$ (K)	$E_{\mathrm{clo}}$ (%)
E-MPINN	IJHMT’25	2.95	0.793	4.59	4.28	9.358	7.26	5.93	7.451	8.04	6.94	10.487	239.33
RoPINN-CHT	NeurIPS’24	5.49	2.366	25.36	4.34	9.265	12.75	5.63	7.394	12.63	10.99	9.600	10.20
CoPINN-CHT	ICML’25	3.16	0.869	5.86	4.53	12.108	12.28	5.84	7.435	5.75	6.97	10.334	4092.33
MUSA-PINN-CHT	ICML’26	1.75	1.117	11.88	4.20	2.108	1.29	3.65	0.937	2.87	5.04	2.326	5.96
Cova-PINN	Ours	1.40	0.316	1.01	2.96	1.890	0.92	3.12	0.828	2.73	2.98	1.006	4.09

Table 1: End-to-end accuracy on four TPMS exchangers. We report full-field temperature, outlet bulk-temperature, and device-level closure errors. Lower is better; bold and underline mark the best and second-best results. IMP. is Cova-PINN’s improvement over the strongest baseline. Exact definitions and heat-duty results are given in Appendix G and H .

Figure 2: TPMS exchanger geometries. Top: Primitive and Gyroid; bottom: Diamond and IWP. Each exchanger contains five streamwise unit cells. Red and blue denote the hot- and cold-fluid domains, while dark gray denotes the separating solid wall.

Figure 3: Outlet-temperature comparison on the Diamond exchanger at $x=5$ . Top: predicted temperatures from the five methods and CFX; bottom: the corresponding absolute-error fields, followed by $\Delta E=|T_{\mathrm{MUSA}}-T_{\mathrm{CFX}}|-|T_{\mathrm{Cova}}-T_{\mathrm{CFX}}|$ in the rightmost panel. Prediction and absolute-error panels use matched color ranges. Positive red values in $\Delta E$ indicate lower Cova-PINN error, whereas negative blue values indicate lower MUSA-PINN-CHT error.

Figure 4: Longitudinal temperature comparison on the Diamond exchanger at $z=0.5$ . From top to bottom: CFX, MUSA-PINN-CHT, and Cova-PINN temperatures, followed by $\Delta E=|T_{\mathrm{MUSA}}-T_{\mathrm{CFX}}|-|T_{\mathrm{Cova}}-T_{\mathrm{CFX}}|$ . The temperature panels share one color range. Positive red values indicate lower Cova-PINN error; negative blue values indicate lower MUSA-PINN-CHT error.

Method	$\boldsymbol{E_{\mathrm{field}}}$ (K)	$\boldsymbol{E_{\mathrm{out}}}$ (K)	$\boldsymbol{E_{Q}}$ (%)	$\boldsymbol{E_{\mathrm{clo}}}$ (%)
MUSA-PINN-CHT	5.996	4.250	7.037	13.575
Cova-PINN	4.802	1.354	1.998	3.590

Table 2: Cross-family evaluation on the geometrically distinct DualMS exchanger. Cova-PINN is compared with MUSA-PINN-CHT, its closest complex-geometry baseline, using field, outlet, heat-duty, and closure errors. Lower is better for all metrics.

Figure 5: Temperature field comparison on DualMS. All panels show the common mid-plane of the two counter-flow U-shaped passages. Top: MUSA-PINN-CHT, Cova-PINN, and CFX temperatures. Bottom: the two absolute-error fields and $\Delta E=E_{\mathrm{MUSA}}-E_{\mathrm{Cova}}$ ; positive red values indicate lower Cova-PINN error, whereas negative blue values indicate lower MUSA-PINN-CHT error.

Variant	$E_{\mathrm{field}}$ (K)	$E_{\mathrm{out}}$ (K)	$E_{\mathrm{clo}}$ (%)
Full Cova-PINN	2.98	1.006	4.09
w/o Local CV (L)	4.81	2.157	6.59
w/o Global Closure (G)	5.00	2.335	7.32
w/o Solid-Mean Reg. (R)	4.56	4.568	23.87

Table 3: Component-wise ablation of Cova-PINN. The controlled study uses the IWP benchmark throughout. Starting from full Cova-PINN, each variant removes local composite-control-volume conservation (L), global paired-wall heat-rate closure (G), or annealed solid-mean regularization (R), while keeping all other training settings fixed. Lower is better for all metrics.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Boundary	Flow condition	Thermal condition
Hot inlet	Uniform, $\|\mathbf{u}_{h}^{\rm d}\|=1\,\mathrm{m\,s^{-1}}$ along $+x$	$T_{h}^{\rm d}=80\,^{\circ}\mathrm{C}$
Cold inlet	Uniform, $\|\mathbf{u}_{c}^{\rm d}\|=1\,\mathrm{m\,s^{-1}}$ along $+x$	$T_{c}^{\rm d}=20\,^{\circ}\mathrm{C}$
Fluid outlets	Mean gauge pressure $\overline{p}_{f}^{\rm d}=0\,\mathrm{Pa}$	No direct thermal constraint
$\Gamma_{hs},\Gamma_{sc}$	No slip	Temperature and normal-flux continuity
External solid boundary	–	Adiabatic

Appendix

Table 4: Boundary conditions for the four TPMS benchmarks. All methods use the same flow and thermal conditions; no direct thermal constraint is imposed at the fluid outlets.

Quantity	Value
$Re_{h}/Re_{c}$	$100/100$
$T_{h,\mathrm{in}}^{\rm d}/T_{c,\mathrm{in}}^{\rm d}$	$80/20\,^{\circ}\mathrm{C}$
$U_{0,h}^{\rm d}/U_{0,c}^{\rm d}$	$1/1\,\mathrm{m\,s^{-1}}$
$\rho_{f}$	$1\,\mathrm{kg\,m^{-3}}$
$c_{p,f}$	$1000\,\mathrm{J\,kg^{-1}\,K^{-1}}$
$k_{f}$	$0.6069\,\mathrm{W\,m^{-1}\,K^{-1}}$

Appendix

Table 5: Shared operating conditions and thermal material parameters for the TPMS benchmarks. Both streams use the listed values in every topology; topology-specific

L_{0}

\mu_{f}

, and

Pe_{f}

are reported in Table 6 .

Geometry	Array	$\boldsymbol{L_{0}}$ (m)	$\boldsymbol{Re_{h}=Re_{c}}$	$\boldsymbol{\mu_{f}}$ (Pa s)	$\boldsymbol{Pe_{f}}$
Primitive (P)	$5\times 1\times 1$	$0.5220$	$100$	$5.220{\times}10^{-3}$	$860.109$
Gyroid (G)	$5\times 1\times 1$	$0.4490$	$100$	$4.490{\times}10^{-3}$	$739.825$
Diamond (D)	$5\times 1\times 1$	$0.3910$	$100$	$3.910{\times}10^{-3}$	$644.258$
IWP	$5\times 1\times 1$	$0.4356$	$100$	$4.356{\times}10^{-3}$	$717.746$

Appendix

Table 6: Topology-specific reference lengths and dimensionless flow parameters for the TPMS benchmarks. Here

L_{0}=4A_{\mathrm{in}}/P_{\mathrm{in}}

. With

\rho_{f}=1\,\mathrm{kg\,m^{-3}}

and

U_{0,f}=1\,\mathrm{m\,s^{-1}}

, the listed viscosity gives

Re_{h}=Re_{c}=100

, and

Pe_{f}=\rho_{f}c_{p,f}U_{0,f}L_{0}/k_{f}

Item	Value
Center support	$\Gamma_{\mathrm{w}}=\Gamma_{hs}\cup\Gamma_{sc}$
Center distribution	Random sampling from solid-wall samples
Fixed radius $r$	$0.05$
Training control volumes $N_{\mathrm{cv}}$	$50$
Candidate points $N_{B}$	$10{,}000$ per ball
Exterior-boundary handling	Retain and integrate reached patches

Appendix

Table 7: Sampling and quadrature configuration for local composite control volumes. The same settings are used in all Cova-PINN experiments; the single fixed radius requires no scale aggregation.

Category	Flow stage	Thermal stage
Input/output	$(x,y,z)\mapsto(u,v,w,p)$	$(x,y,z)\mapsto T_{m}$
Networks	Two, one per fluid	Three, one per medium
Architecture	$3{\to}60$ Fourier features, then $5{\times}256$ $\tanh$ layers	$3{\to}60$ Fourier features, then $5{\times}256$ $\tanh$ layers
Input encoding	Linear Fourier frequencies in $[1,2.5]$ ( Tancik et al. 2020 )	Linear Fourier frequencies in $[1,2.5]$ ( Tancik et al. 2020 )
Initialization	Xavier-normal weights; zero biases	Xavier-normal weights; zero biases
Optimizer	SOAP ( Wang et al. 2025 ) , default parameters	SOAP ( Wang et al. 2025 ) , default parameters

Appendix

Table 8: Cova-PINN network, optimization, and conservation-quadrature configuration. The flow networks are trained first and frozen before thermal optimization; all subnetworks use the same Fourier-feature MLP backbone.

Stage/objective	Weight(s)	Value(s)
Flow strong continuity and momentum	$\lambda_{\mathrm{div}},\lambda_{\mathrm{mom}}$	$1,\ 0.1$
Flow inlet, outlet, and no-slip wall	$\lambda_{\mathrm{in}}^{u},\lambda_{\mathrm{out}}^{u},\lambda_{\mathrm{w}}^{u}$	$10,\ 10,\ 10$
MUSA weak continuity (large/medium/small)	$\lambda_{\mathrm{wk,div}}^{L/M/S}$	$100,\ 100,\ 100$
MUSA weak momentum (large/medium/small)	$\lambda_{\mathrm{wk,mom}}^{L/M/S}$	$4,\ 25,\ 100$
Thermal PDE and boundary conditions	$\lambda_{\mathrm{PDE}},\lambda_{\mathrm{bc}}$	$1,\ 1$
Interface temperature and heat flux	$\lambda_{\Gamma,T},\lambda_{\Gamma,q}$	$1,\ 1$

Appendix

Table 9: Loss weights used by Cova-PINN and the baseline objectives. Strong-form weights are shared across methods; MUSA weak-form flow weights are shared by MUSA-PINN-CHT and Cova-PINN, while the local, global, and regularization weights apply to Cova-PINN.

Method	P	G	D	IWP
E-MPINN	3.03	3.76	13.09	58.15
RoPINN-CHT	16.12	6.89	13.81	14.43
CoPINN-CHT	3.79	6.61	12.96	167.03
MUSA-PINN-CHT	7.95	3.14	1.63	5.74
Cova-PINN	1.57	2.92	1.49	4.21

Appendix

Table 10: Reference-normalized two-sided heat-duty error

E_{Q}

(%) on four TPMS exchangers. Lower is better; bold and underline denote the best and second-best results. Each value evaluates the complete end-to-end model produced by the corresponding training protocol.

Figure 6: Full longitudinal comparison on the Diamond exchanger at $z=0.5$ . The left column shows the predicted temperature fields of all methods and the CFX reference; the right column shows the corresponding pointwise absolute-error fields. Temperature and absolute-error panels use matched color ranges. The final panel reports $\Delta E=|T_{\mathrm{MUSA}}-T_{\mathrm{CFX}}|-|T_{\mathrm{Cova}}-T_{\mathrm{CFX}}|$ ; positive red values indicate lower Cova-PINN error, whereas negative blue values indicate lower MUSA-PINN-CHT error.

Method	Training stage	Updates	Time (h)
E-MPINN	Joint	$10{,}000$	$20.43$
RoPINN-CHT	Joint	$10{,}000$	$27.87$
CoPINN-CHT	Joint	$10{,}000$	$21.85$
MUSA-PINN-CHT	Joint	$10{,}000$	$32.76$
Cova-PINN (flow)	Flow stage	$5{,}000$	$25.12$
Cova-PINN (thermal)	Thermal stage	$5{,}000$	$16.18$

Appendix

Table 11: End-to-end wall-clock cost under a common

10{,}000

-update schedule. All runs use one NVIDIA A40 GPU. Baselines train jointly, whereas Cova-PINN reports its sequential flow and thermal stages together with their total.

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MUSA-PINN: Multi-scale Weak-form Physics-Informed Neural Networks for Fluid Flow in Complex Geometries

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Neural PDE Solvers PDE Solving Physics-Informed ML Computational Fluid Dynamics

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While Physics-Informed Neural Networks (PINNs) offer a mesh-free approach to solving fluid-flow PDEs, standard point-wise residual minimization suffers from convergence pathologies in topologically complex domains like Triply Periodic Minimal Surfaces (TPMS). The locality bias of point-wise constraints fails to propagate global information through tortuous channels, causing unstable gradients and conservation violations. To address this, we propose the Multi-scale Weak-form PINN (MUSA-PINN), which reformulates Navier-Stokes equation constraints as integral conservation laws over hierarchical spherical control volumes. We enforce continuity and momentum conservation via flux-balance residuals on control surfaces. Our method utilizes a three-scale subdomain strategy-comprising large volumes for long-range coupling, skeleton-aware meso-scale volumes aligned with transport pathways, and small volumes for local refinement-alongside a two-stage training schedule prioritizing continuity. Experiments on steady incompressible flow in TPMS geometries show MUSA-PINN outperforms state-of-the-art baselines, reducing relative errors by up to 93% and preserving mass conservation.

Abstract

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