cs.LGOct 4, 2026

Component-Level Evaluation of Adaptive PINN Training for CFD-Oriented Crystal Growth Simulation

Authors: Niruta Chapagain, Rohit Raj, Bertwin Kurisinkal Shine, Aditya A S

Organizations: University of Europe for Applied Sciences Potsdam, Germany · Vienna, Austria · Technische Universität Chemnitz Chemnitz, Germany · quasi digital Berlin, Germany

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Abstract

Physics-informed neural network (PINN) training minimizes a weighted combination of partial differential equation (PDE), boundary-condition, and initial-condition losses. Because adaptive methods modify these weights during training, their weighted total losses are not always directly comparable. We compare fixed-weight PINN, gradient-normalized PINN (GNPINN), and a rule-based adaptive controller (AgenticPINN) under matched settings on a heat-equation benchmark and a simplified Czochralski-oriented thermal-fluid problem. In the crystal-growth MLP experiment, adaptive control reduced the PDE residual from the order of 10−510^{-5} to 10−610^{-6}, while the boundary-condition loss increased from the order of 10−510^{-5} to 10−210^{-2}. On the heat-equation benchmark, GNPINN achieved the lowest relative L2L_2 field error (0.054), whereas AgenticPINN obtained the smallest PDE residual but a relative L2L_2 error of 1.368. Gaussian-process surrogates were additionally evaluated using case-wise holdout tests on corrected Czochralski CFD parameter sweeps. The temperature-field error for the temperature sweep was approximately 6%, whereas the axial-velocity error for the crystal-rotation sweep was approximately 42%. These findings show that adaptive control can improve equation satisfaction while weakening other physical constraints. PINN training should therefore be evaluated using separate PDE, boundary-condition, and solution-error metrics rather than weighted total loss alone.

Figures & tables

ProblemTrainerPDE lossBC/IC lossWeighted totalRelative L2L_{2} error
Heat-equation MLPBaseline PINN3.96×10−33.96\times 10^{-3}1.15×10−31.15\times 10^{-3}5.11×10−35.11\times 10^{-3}8.43×10−28.43\times 10^{-2}
Heat-equation MLPGNPINN9.74×10−39.74\times 10^{-3}9.05×10−4\mathbf{9.05\times 10^{-4}}1.06×10−21.06\times 10^{-2}5.41×10−2\mathbf{5.41\times 10^{-2}}
Heat-equation MLPAgenticPINN8.22×10−7\mathbf{8.22\times 10^{-7}}2.00×10−12.00\times 10^{-1}2.08×10−32.08\times 10^{-3}1.3681.368
Crystal-growth MLPBaseline PINN9.26×10−69.26\times 10^{-6}1.63×10−51.63\times 10^{-5}2.55×10−52.55\times 10^{-5}–
Crystal-growth MLPGNPINN1.61×10−51.61\times 10^{-5}1.01×10−41.01\times 10^{-4}1.17×10−41.17\times 10^{-4}–
Crystal-growth MLPAgenticPINN1.14×10−6\mathbf{1.14\times 10^{-6}}1.29×10−21.29\times 10^{-2}2.43×10−42.43\times 10^{-4}–
Table 1: Final PINN losses under matched settings. PDE and BC/IC losses are reported separately because the adaptive strategy changes their relative weighting during training. Relative L2L_{2} error is available only for the heat-equation benchmark.
Figure 1: Component-level PINN and CFD-surrogate results. left: Final PDE and constraint losses for the simplified crystal-growth PINN, shown on a logarithmic scale. AgenticPINN obtains the smallest PDE residual but the largest constraint loss. right: Temperature-field normalized RMSE for case-wise holdouts from the three CFD parameter sweeps.
Figure 2: Adaptive loss-weight dynamics for the simplified crystal-growth (left) and heat-equation (right) PINNs. In both experiments, λPDE\lambda_{\mathrm{PDE}} increases to its upper clipping bound while the constraint weight decreases to its lower bound, producing a 10410^{4} disparity between the two loss weights.
Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Analytical heat-equation solution and temperature predictions from the Baseline PINN, GNPINN, and AgenticPINN MLP runs. On the common evaluation grid, the corresponding relative L2L_{2} errors are 8.43×10−28.43\times 10^{-2} , 5.41×10−25.41\times 10^{-2} , and 1.3681.368 . All four panels use the same axes and color scale. Although AgenticPINN has the smallest PDE residual, its larger field error is consistent with its substantially larger boundary and initial-condition loss.
Figure 4: Post-processed velocity, pressure, and temperature predictions at t=0.20t=0.20 for the simplified crystal-growth MLP experiment. Rows correspond to Baseline PINN, GNPINN, and AgenticPINN. The methods produce noticeably different velocity and pressure structures, while their temperature fields are more similar. These panels support the component-level interpretation of the training results but do not constitute validation against an industrial CFD solution.

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