Domain-informed Adaptive Sampling for Generalizable PINNs in Metal Additive Manufacturing via Conditional Flow Matching

Hyeonsu Lee; Jihoon Jeong

cs.LGOct 6, 2026

Domain-informed Adaptive Sampling for Generalizable PINNs in Metal Additive Manufacturing via Conditional Flow Matching

Authors: Hyeonsu Lee, Jihoon Jeong

Organizations: Wm Michael Barnes ’64 Department of Industrial and Systems Engineering Texas A&M University College Station, TX 77843, USA

Abstract

Accurate thermal modeling is essential in metal additive manufacturing (AM) for understanding the process-structure-property chain. Physics-informed neural networks (PINNs) offer effective surrogate thermal modeling by minimizing physics-based residual losses at collocation points. However, prior works typically rely on manually-crafted, static collocation sampling strategies, which are neither principled nor scalable across process conditions, hindering their generalization capability. In this work, we provide theoretical analysis through empirical risk minimization, showing that process condition-aware adaptive sampling is strictly more favorable than conventional static sampling for generalization. Building on this insight, we propose an adaptive sampling strategy within a two-stage framework: (1) a conditional Flow Matching model that learns approximate high-residual distributions across different process conditions, and (2) a mixed sampling strategy combining this distribution with a domain-informed base distribution to generate adaptive collocation points for refining the PINN predictor. Experiments on metal AM numerical benchmarks demonstrate that our method consistently outperforms state-of-the-art PINN baselines, achieving an average 62.1% reduction in relative $L_2$ error under an identical collocation budget, by capturing process-dependent heat dissipation regions often overlooked in the literature. To the authors' knowledge, this is the first adaptive sampling strategy for PINNs in metal AM, contributing to the enhanced generalization and broader applicability.

Figures & tables

Figure 1 : Visual overview of Section 2.3 . Left: Proposition 2.1 decomposes the generalization error into an optimization error, which is independent of the proposal $q$ , and a statistical error that scales as $\mathcal{O}(\sqrt{\bar{\sigma}^{2}_{q}/N})$ . This implies that reducing the variance of the empirical loss estimator, $\bar{\sigma}^{2}_{q}$ , is a more effective lever for minimizing the statistical error, and thereby the overall generalization error, than merely increasing the total sample size $N$ . Right: To reduce $\bar{\sigma}^{2}_{q}$ , Theorem 2.1 states that conventional $\mu$ -independent static sampling is strictly inferior to adaptive sampling, as its infimum variance $V^{\star}_{\mathrm{nc}}(\theta)$ imposes a hard ceiling, whereas the $\mu$ -conditioned adaptive proposal strictly lowers this variance floor. Lemma 2.1 further characterizes the optimal adaptive proposal as being proportional to the squared residual.

Figure 2 : Domain-informed adaptive collocation via CFM. Top (a–d): CFM transport of uniform base samples ( $\tau=0$ ) into the adaptive proposal $q^{\star}_{\theta_{\mathrm{src}}}(\cdot\mid t,\mu)$ ( $\tau=1$ ), which concentrates in high-residual regions. Bottom : (e) Domain-informed base $b(\cdot\mid t)$ ; (f–h) Training mixture $q_{\alpha}^{b}$ at $t=0.5,1.5$ , and $2.5\,\text{s}$ , where the base remains fixed while the CFM component tracks high-residual regions (visualized for Ti-6Al-4V).

Figure 3 : Ground-truth temperature field snapshots at $t=0.5$ , $1.5$ , and $2.5\,\mathrm{s}$ obtained via high-fidelity JAX-AM simulation for the bare-plate scanning benchmark (visualized using Ti–6Al–4V). The thermal evolution captures the localized moving laser heat source and subsequent heat diffusion.

		In-distribution (ID)			Out-of-distribution (OOD)
Method	Average	Ti-6Al-4V	Inconel 718	SS 316L	AlSi10Mg	Copper
		$(4430,560,6.7)$	$(8220,435,11.4)$	$(8000,500,16.0)$	$(2670,950,150)$	$(8960,385,401)$
N-PINN	$4.82$	$6.19\pm 1.00$	$6.09\pm 0.83$	$5.94\pm 0.63$	$4.25\pm 0.35$	$1.63\pm 0.13$
AP-PINN	$3.91$	$5.80\pm 0.32$	$4.85\pm 0.30$	$4.85\pm 0.28$	$2.86\pm 0.07$	$1.18\pm 0.05$
Proposed	$\mathbf{1.48}$	$\mathbf{2.55\pm 0.49}$	$\mathbf{1.71\pm 0.36}$	$\mathbf{1.45\pm 0.21}$	$\mathbf{1.19\pm 0.21}$	$\mathbf{0.51\pm 0.10}$
Reduction [%]	$\mathbf{62.1\%}$	$\mathbf{56.0\%}$	$\mathbf{64.7\%}$	$\mathbf{70.1\%}$	$\mathbf{58.4\%}$	$\mathbf{56.8\%}$

Table 1 : Relative

L_{2}

error [%] (mean

\pm

std over five seeds) across five materials. Average denotes the mean across all five materials. The last row reports the error reduction of the proposed method relative to the AP-PINN baseline. Lower is better.

Figure 4 : Training dynamics (mean $\pm$ std over five seeds): (a) average relative $L_{2}$ error on the five materials, and (b–d) the PDE, BC, and IC training losses evaluated on each method’s own sampled collocation points. The proposed method reaches a lower relative $L_{2}$ error despite higher training losses on the three losses, which is a direct consequence of concentrating collocation points in high-residual regions.

Figure 5 : Comparison of empirical loss $\widehat{\mathcal{L}}_{N}(\theta;q)$ ( 3 ) and population loss $\mathcal{L}(\theta)$ ( 2 ) (mean $\pm$ std over five seeds): (a) training evolution and (b) final values. The proposed method exhibits a higher empirical loss during training, yet consistently achieves a lower population loss by focusing samples on high-residual regions, which leads to more effective learning.

Panel of Figure 6 — Figure 6 : Comparison of thermal histories at increasing depths below the laser scan track across five materials (columns) at the top surface, $2\,\mathrm{mm}$ , and $4\,\mathrm{mm}$ depth (rows) (mean $\pm$ std over five seeds): (a) predicted temperature profiles and FEM reference and (b) temperature deviations $\Delta T=T_{\theta}-T$ relative to the FEM reference. Both methods match well at the top surface under direct laser scanning, but the AP-PINN baseline underperforms significantly in late-time and far-field regions where heat dissipation governs over laser irradiation. Ultimately, AP-PINN misses the subsurface thermal profiles entirely at $4\,\mathrm{mm}$ depth across all materials, whereas the proposed method accurately captures it.

Panel of Figure 7 — Figure 7 : Temperature deviation $\Delta T=T_{\theta}-T$ at $t=0.5$ , $1.5$ and $2.5$ s (rows), with each method’s collocation points overlaid. Top-surface panels show the top boundary, cross-section panels the domain group; the dashed line marks the laser center. Both the total number of collocation points are identical while AP-PINN draws them from $b(x\mid t)$ whereas the proposed method from the mixture $q_{\alpha}^{b}$ in ( 10 ) with $\alpha=0.6$ . The collocation points drawn in AP-PINN are concentrated nearby the laser irradiation zone and the top surface, which results in significant discrepancy from the FEM reference at heat dissipation regions. In contrast, the proposed method effectively allocates the collocation points on both the laser irradiation zone and the heat dissipation regions at a given time step, resulting in considerable attenuation of the discrepancies shown in the AP-PINN baseline.

Figure 8 : Marginal distributions of collocation points along each spatial axis, averaged over $t$ , across the five evaluation materials.

Figure 9 : Relative $L_{2}$ error (mean $\pm$ std over five seeds) across the mixture weight $\alpha$ for the two base distributions.

Panel of Figure 10 — Figure 10 : Signed deviation $\Delta T=T_{\text{PINN}}-T_{\text{FEM}}$ at $t=0.5$ , $1.5$ , $2.5$ s for models trained with the domain-informed and the uniform base at the same $\alpha=0.6$ , with each configuration’s collocation points overlaid.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 11 : Temperature field at $t=0.5$ s for the five validation materials (rows): FEM reference and the proposed method, on the top surface and on the $y=5$ mm cross-section. Note that each material uses its own color scale due to differences in thermophysical properties.

Figure 12 : Temperature field at $t=1.5$ s (same layout and color scales as Figure 11 ). The laser has advanced to $x=20$ mm and the trailing heat tail has developed.

Figure 13 : Temperature field at $t=2.5$ s (same layout and color scales as Figure 11 ). The proposed method reproduces the melt pool, the trailing heat tail, and the depth of heat penetration for every material, including the out-of-distribution AlSi10Mg and Copper.

Figure 14 : Signed deviation and collocation overlay for Ti-6Al-4V ( $k_{c}=6.7$ W m -1 K -1 ), plotted as in Fig. 7 .

Figure 15 : Signed deviation and collocation overlay for SS316L ( $k_{c}=16$ W m -1 K -1 ), plotted as in Fig. 7 .

Figure 16 : Signed deviation and collocation overlay for AlSi10Mg ( $k_{c}=150$ W m -1 K -1 ), plotted as in Fig. 7 .

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Apr 16, 2026, cs.LGJ/K move · Enter open · S save

Material-Agnostic Zero-Shot Thermal Inference for Metal Additive Manufacturing via a Parametric PINN Framework

Hyeonsu Lee, Jihoon Jeong

Wm Michael Barnes ’64 Department of Industrial and Systems Engineering Texas A&M University College Station, TX 77843, USA

Parametric Physics-Informed Neural Network Zero-Shot

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Accurate thermal modeling in metal additive manufacturing (AM) is essential for understanding the process-structure-performance relationship. While prior studies have explored generalization across unseen process conditions, they often require extensive datasets, costly retraining, or pre-training. Generalization across different materials also remains relatively unexplored due to the challenges posed by distinct material-dependent thermal behaviors. This paper introduces a parametric physics-informed neural network (PINN) framework for zero-shot generalization across arbitrary materials without labeled data, retraining, or pre-training. The framework adopts a decoupled parametric PINN architecture that separately encodes material properties and spatiotemporal coordinates, fusing them through conditional modulation to better align with the multiplicative role of material parameters in the governing equation and boundary conditions. Physics-guided output scaling derived from Rosenthal's analytical solution and a hybrid optimization strategy are further incorporated to enhance physical consistency, training stability, and convergence. Experiments on bare plate laser powder bed fusion (LPBF) across diverse metal alloys, including both in-distribution and out-of-distribution cases, demonstrate effective zero-shot generalizability along with superior training efficiency. Specifically, the proposed framework achieved up to a 64.2% reduction in relative L2 error compared to the non-parametric baseline while surpassing its performance within only 4.4% of the baseline training epochs. Ablation studies confirm that the proposed framework's components are broadly applicable to other PINN-based approaches. Overall, the proposed framework provides an efficient and scalable material-agnostic solution for zero-shot thermal modeling, contributing to more flexible and practical deployment in metal AM.

Abstract

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