PaperlayerView exact revision on arXiv

Source-linked AI summary

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

arXiv:2610.11108v1cs.LGphysics.flu-dyn

TL;DR

Separate domain supports in multi-domain PINNs can yield plausible temperature fields but inaccurate end-to-end CHT transfer. Cova-PINN aligns local composite-control-volume balances and global paired-wall closure with the thermal interaction path. Against MUSA-PINN-CHT, it reduces average outlet-temperature and closure errors across four TPMS topologies, with gains also reported on DualMS.

  • Problem

    Separately sampled domain and interface residuals do not directly constrain complete-path transfer, so pointwise minimization does not guarantee device-level energy conservation.

    Source: §1 · p. 1, §3.3 · p. 4

  • Method

    Cova-PINN jointly optimizes cross-domain local control-volume balances and global paired-wall closure, with annealed solid-mean regularization during early training.

    Source: §1 · p. 1, §4.1 · p. 4

  • Results

    37.7% lower average outlet-temperature error and 60.2% lower device-level closure error across four TPMS topologies relative to MUSA-PINN-CHT, with field, heat-duty, and closure gains also reported on DualMS.

    Source: §6 · p. 7

  • Takeaways & Limitations

    Across the tested TPMS and geometrically distinct DualMS exchangers, conservation-support alignment improves field, outlet, heat-duty, and closure accuracy relative to the closest baseline.

    Source: §1 · p. 1, §6 · p. 7

    The study assumes steady, constant-property CHT with temperature-independent flow fields; transient transport, temperature-dependent flow–thermal feedback, and the added quadrature training cost remain outside or constrain its scope.

    Source: §3.2 · p. 3, §6 · p. 7

Abstract

from arXiv · show

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.

1 Introduction

Separate domain supports can leave multi-domain PINNs with plausible temperature fields but inaccurate exchanger-scale transfer. Cova-PINN aligns conservation objectives with the hot-fluid–solid–cold-fluid path and reduces errors across TPMS topologies and DualMS.

1 Introduction

  • Separately supported domain losses can produce plausible fields yet fail to ensure globally consistent transfer along a conserved quantity crossing learned domains.For steady CHT without internal heat generation, heat released by the hot stream must equal heat absorbed by the cold stream.Source: §1 · p. 1
  • Prior methods address domain decomposition, CHT training, pointwise optimization, and internal-flow conservation, but do not align conservation support with the complete thermal interaction path.The design question is where an integral constraint is supported relative to that interaction topology.Source: §1 · p. 1
  • Cova-PINN jointly optimizes wall-centered local composite-control-volume balances, global paired-wall closure, and an annealed solid-mean regularizer.Its two MUSA-PINN flow networks are trained independently and frozen before joint optimization of the three temperature networks.Source: §1 · p. 1, Figure 1 · p. 2
  • Cova-PINN reduces field, outlet, heat-duty, and closure errors across TPMS topologies and DualMS, with ablations supporting all three components.The abstract identifies MUSA-PINN-CHT as the closest baseline and reports consistent gains on DualMS.Source: §1 · p. 1

2 Related Work

Prior work advances multi-network representations, weak and integral physics supervision, and PINN optimization in complex geometries. Cova-PINN combines these directions by aligning conservation support with the multi-domain thermal interaction path.

2 Related Work

  • Multi-domain PINNs couple networks across decomposed supports or material regions; E-MPINN is selected as the task-specific multi-network CHT baseline.The related work also describes MDPINN-GD's global inlet–outlet mass-flow conservation and prior thermal models for heterogeneous layers and interfaces.Source: §2 · p. 2
  • Weak-form and control-volume methods supervise finite-region physics, but prior constraints typically remain within one PDE domain or a predefined geometric decomposition.Cova-PINN instead places composite supports across adjoining materials along the hot-fluid–solid–cold-fluid path and adds device-scale paired-wall closure.Source: §2 · p. 2
  • RoPINN and CoPINN improve local-objective sampling or optimization, while MUSA-PINN places multiscale weak-form control volumes along complex flow pathways.MUSA-PINN is used as the complex-geometry flow baseline and closest structural comparator; Cova-PINN extends conservation supervision to the coupled thermal path.Source: §2 · pp. 2–3

3 Problem Formulation

The formulation represents steady CHT as energy transfer across separately learned hot-fluid, solid, and cold-fluid domains. Because the standard pointwise objective uses separate supports, it does not by itself guarantee device-level energy conservation.

3.1 Physical Setting and Energy Transfer

  • Heat travels from the hot-fluid domain across Γhs, through the solid, across Γsc, and into the cold-fluid domain under steady forced convection with constant properties and no internal heat generation.The setup also assumes negligible buoyancy and the same fluid in both streams.Source: §3.1 · p. 3
  • Dimensionless temperatures scale the hot and cold inlets to 1 and 0; fluid energy flux combines advection and conduction, while solid flux represents conduction.All three fluxes share a common scale so their surface integrals can be added in cross-domain control volumes.Source: §3.1 · p. 3
  • At Γhs and Γsc, the interface conditions impose no slip, temperature continuity, and normal heat-flux continuity.The energy equations describe convection and diffusion in each fluid and conduction in the solid.Source: §3.1 · p. 3

3.2 Sequential Multi-Network Learning Setting

  • Two flow fields and three temperature fields use independent networks; flow networks are trained first, then frozen, leaving a thermal problem with fixed advection.This is sequential CHT rather than bidirectionally coupled thermofluid learning.Source: §3.2 · p. 3
  • The common pointwise thermal objective penalizes energy equations, boundary conditions, and interface transmission using separately averaged domain- and boundary-specific samples.No individual residual directly spans the complete transfer path.Source: §3.2 · p. 3

3.3 End-to-End Conservation Requirement

  • For the steady exchanger with an adiabatic exterior solid boundary and no internal generation, hot-stream heat release must equal cold-stream heat absorption.Inlet–outlet rates Qh and Qc are normalized by Q0 for device-level evaluation, whereas solid-side wall integrals form the global training objective.Source: §3.3 · pp. 3–4
  • Separate finite-sample supports do not explicitly optimize the integrated cross-domain relation, so minimizing pointwise residuals alone does not guarantee device-level conservation.The resulting cross-domain conservation gap can appear as inaccurate outlet bulk temperatures and nonzero Gglobal.Source: §3.3 · p. 4

4 Cova-PINN

Cova-PINN co-optimizes local cross-domain balances and global paired-wall closure alongside standard pointwise CHT physics. An annealed solid-mean regularizer supports early training without prescribing the solid temperature's spatial structure.

4.1 Overview and Global–Local Co-Optimization

  • After freezing the independently trained flow networks, Cova-PINN jointly updates temperature networks using pointwise CHT physics and coordinated local–global conservation objectives.The solid-mean regularizer anchors early training, then its weight vanishes so the final solution is governed by CHT and conservation objectives.Source: §4.1 · p. 4

4.2 Local Cross-Domain Composite Control Volumes

  • Wall-centered spherical supports intersect adjoining material regions, forming composite control volumes whose material-labeled outer-patch fluxes define local balances.Each residual aggregates flux contributions from all intersected materials and jointly updates their corresponding temperature networks.Source: §4.2 · p. 4

4.3 Global Paired-Wall Heat-Rate Closure

  • Global closure pairs the hot-side and cold-side walls and evaluates both heat rates from solid-side fluxes, imposing the steady balance between rates.The closure loss updates the solid network, while pointwise normal-flux conditions transmit its effect to fluid temperature networks; inlet–outlet Gglobal remains an independent diagnostic.Source: §4.3 · p. 4

4.4 Annealed Solid-Mean Temperature Regularization

  • The regularizer anchors the solid-domain mean temperature during early optimization without prescribing its spatial structure, then decays linearly to zero over the first half of training.This zero-order regularizer preserves spatial degrees of freedom in Ts and influences fluid networks only through coupled physics losses.Source: §4.4 · p. 5

5 Experiments

The experiments compare Cova-PINN with CHT-specific, optimization-oriented, and geometry-aware baselines across four TPMS exchangers and the geometrically distinct DualMS design. Results report reduced errors across both geometry families, while ablations show complementary contributions from the framework’s components.

5.1 Experimental Setup

  • Four TPMS exchangers and geometrically distinct DualMS were evaluated against CHT-specific, optimization-oriented, and geometry-aware baselines with matched capacity, CHT physics, and update counts, using field, outlet, and closure errors.The TPMS cases vary in channel connectivity and fluid–solid interface orientation; both streams operate at Re = 100.Source: §5.1 · p. 5

5.2 Comparative Analysis on TPMS Exchangers

  • 28.6%, 37.7%, 60.2%, and 44.8%: relative to MUSA-PINN-CHT, Cova-PINN reduces average Efield, Eout, Eclo, and EQ, respectively, across four TPMS exchangers.Table 1 reports end-to-end accuracy across the four exchangers; lower errors are better.Source: §5.2 · pp. 5–6, Table 1 · p. 6

5.3 Cross-Family Evaluation on DualMS

  • 19.9%, 68.1%, 71.6%, and 73.6%: on DualMS, Cova-PINN reduces Efield, Eout, EQ, and Eclo, respectively, versus MUSA-PINN-CHT.DualMS has separate U-shaped passages carrying the hot and cold streams in opposite directions.Source: Table 2 · p. 6, §5.3 · p. 6

5.4 Ablation Studies

  • On IWP, every component removal degrades all three reported metrics; removing G gives the largest field error (5.00 K), while removing R most harms outlet and closure accuracy (4.568 K and 23.87%).EQ rises from 4.21% to 4.36%, 4.75%, and 12.71% without L, G, and R, respectively; the distinct degradation patterns indicate complementary contributions.Source: Table 3 · p. 7, §5.4 · p. 7

6 Discussion and Conclusion

Relative to MUSA-PINN-CHT, Cova-PINN reduces average errors across four TPMS topologies and also reports gains on DualMS; the study is limited to steady, constant-property CHT with temperature-independent flow fields.

6 Discussion and Conclusion

  • Across four TPMS topologies, Cova-PINN reduces average field, outlet, heat-duty, and closure errors by 28.6%, 37.7%, 44.8%, and 60.2%, respectively, versus MUSA-PINN-CHT.On DualMS, the corresponding reductions are 19.9%, 68.1%, 71.6%, and 73.6%.Source: §6 · p. 7
  • The study covers steady, constant-property CHT with temperature-independent flow fields; transient transport and temperature-dependent flow–thermal feedback remain outside its scope.Local-control-volume and paired-wall quadrature also increase training cost.Source: §6 · p. 7
  • Future work proposes extending support-aligned conservation to coupled and transient thermofluid problems, adaptive control-volume placement, sparse quadrature, and parameterized transferability studies.Source: §6 · p. 7

B Geometry Construction; C Spherical Composite Control Volumes; E Reference Solutions

F Experimental Configuration

The experimental-configuration material is organized around network and loss setup, sequential training, baseline selection, and fairness.

F Experimental Configuration

H Complete Results

The complete-results section lists heat-duty accuracy and longitudinal field comparison analyses.

H.1 Reference-Normalized Heat-Duty Accuracy

H.2 Full Longitudinal Field Comparison

A Complete Problem Specification

The problem specification sets steady three-domain CHT assumptions, governing and interface conditions, and benchmark boundary configurations.

A.1 Domains, Assumptions, and Nondimensionalization

  • The setup assumes steady incompressible forced convection, constant properties, negligible buoyancy, and no volumetric heat sources; flow fields are learned first and frozen during thermal training.The nondimensionalization uses a geometric scale and inlet mean speed; the characteristic length is the inlet hydraulic diameter.Source: §A.1 · p. 11

A.2 Governing Equations and Interface Conditions

  • The three media use energy-flux definitions on a common reference heat-flux scale, with conjugate interfaces enforcing no slip, temperature continuity, and flux continuity.Source: §A.2 · p. 11

A.3 Boundary and Material Configuration

  • All methods share the TPMS boundary conditions, and no direct thermal constraint is imposed at fluid outlets.The TPMS cases use Re_h = Re_c = 100 and a 60 K inlet temperature difference; DualMS applies the same inlet, outlet, interface, and external adiabatic conditions to its labeled ports and walls.Source: Table 4 · p. 11, §A.3 · p. 11

B.1 TPMS Benchmark Family

The study benchmarks four TPMS exchanger geometries and a distinct DualMS design, then builds Cova-PINN around local composite-volume balances and global paired-wall closure. Its setup combines area-weighted quadrature, sequential training, and comparisons under a shared update budget.

B.1 TPMS Benchmark Family

  • Primitive, Gyroid, Diamond, and IWP form the five-unit-cell TPMS benchmark family, with common normalized wall thickness and parallel flow.The experiments treat the geometries as distinct benchmark domains rather than comparing their intrinsic heat-exchanger performance.Source: §B.1 · p. 12

B.2 DualMS Geometry

  • DualMS adds a geometrically distinct U-shaped case, using a fixed design without geometry optimization.The watertight solid and two fluid volumes are used directly for sampling and CFX meshing.Source: §B.2 · p. 12

C.1 Composite Outer-Boundary Construction; C.2 Area-Uniform Sampling and Total-Area Weighting

  • Composite control volumes are centered on the conjugate walls and extend across adjoining materials, with area-weighted boundary sampling.The implementation samples 50 centers per refresh, uses radius 0.05 against nominal wall thickness 0.03, and regenerates centers and candidate samples every 500 thermal updates.Source: §C.1 · p. 12, §C.2 · pp. 12–13

C.3 Interface Cancellation

  • Summing material-wise balances cancels interface terms for exact CHT solutions, leaving an outer-boundary residual for local training.The local loss does not directly evaluate interface-flux jumps, so pointwise temperature- and flux-continuity penalties remain necessary.Source: §C.3 · p. 13

D Paired-Wall Heat-Rate Estimator

  • Global closure compares hot- and cold-side heat rates estimated from the solid network using area-uniform interface samples.The paired-wall estimators depend directly only on the solid temperature network and use 120,000 samples across both interfaces.Source: D Paired-Wall Heat-Rate Estimator · p. 13

E.1 Reference Solver

  • ANSYS CFX 2023 R1 supplies reference fields for post-training evaluation, not network optimization or training decisions.The coupled finite-volume solver uses unstructured tetrahedral meshes and continues until RMS equation residuals fall below 10^-8.Source: §E.1 · p. 13

F.1 Networks, Base Losses, and Sequential Training

  • Cova-PINN trains flow networks first, freezes them, and then updates only the three temperature-network parameter sets.The flow and thermal objectives are optimized sequentially rather than as one simultaneous loss.Source: §F.1 · p. 14

F.2 Baseline Selection and Fairness

  • The comparison gives methods 10,000 total updates, matching optimization-update count rather than wall-clock time; baselines jointly optimize flow and temperature networks for the full schedule.Cova-PINN uses 5,000 flow-stage updates followed by 5,000 thermal-stage updates, with learned flow fields frozen during thermal optimization.Source: §F.2 · p. 15

G Evaluation Metrics

The evaluation separates temperature-field and outlet-temperature errors from heat-duty accuracy and device-level closure. It also reports complete TPMS heat-duty results, longitudinal field comparisons, and training cost.

G.1 Temperature-Field and Outlet Errors

  • The pooled full-field temperature error counts every valid evaluation point once across all three media.The outlet metric averages the hot- and cold-stream bulk-temperature errors using the same reference velocity field for prediction and reference estimators.Source: §G.1 · p. 15

H Complete Results; H.1 Reference-Normalized Heat-Duty Accuracy

  • Table 10 reports reference-normalized two-sided heat-duty error for each method–topology pair across four TPMS exchangers.Lower error is better; the table marks the best and second-best results.Source: Table 10 · p. 16

G.2 Heat-Rate Accuracy and Device Closure

  • Reference-normalized two-sided heat-duty error measures whether transferred heat on both streams agrees with CFX, unlike device closure.Outlet-based heat-duty and closure diagnostics are evaluated after the fixed training budget and are not included in training.Source: §H.1 · p. 16, §G.2 · p. 16

H.2 Full Longitudinal Field Comparison

  • Figure 6 compares predicted temperature and pointwise absolute-error fields across Diamond's five streamwise unit cells.The temperature and error panels use matched color ranges.Source: §H.2 · p. 16
  • Wall-clock cost is reported for the fixed 10,000-update schedule on one NVIDIA A40 GPU.The comparison matches total update count, not elapsed time or updates per individual field.Source: Table 11 · p. 16

I Computational Cost

Training cost is measured under a fixed update schedule, not by time to convergence. The supplied visual passage describes a separate Diamond field comparison.

I Computational Cost

  • Wall-clock measurements use one NVIDIA A40 GPU and a fixed 10,000-update schedule.The comparison matches update count rather than wall-clock time; Cova-PINN's added cost reflects staged optimization and its local-control-volume and paired-wall quadrature.Source: I Computational Cost · p. 16
  • Figure 6 compares Diamond temperature fields, pointwise absolute errors, and a signed error-difference panel at z = 0.5.Positive red values indicate lower Cova-PINN error, while negative blue values indicate lower MUSA-PINN-CHT error.Source: Figure 6 · p. 17

Related papers

Loading 2610.11108v1…