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

arXiv:2610.09126 · cs.LG · Submitted 2026-10-06 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Domain-informed Adaptive Sampling for Generalizable PINNs in Metal Additive Manufacturing via Conditional Flow Matching".

Jane: The gist: This work presents, to the best of our knowledge, the first adaptive collocation sampling strategy for PINNs in metal AM.

Tom: First, who's behind it and why it matters.

Paper summary: Tom: So, to wrap up, the paper "Domain-informed Adaptive Sampling for Generalizable PINNs in Metal Additive Manufacturing via Conditional Flow Matching" is arguing that process condition-aware adaptive sampling is critical for making parametric PINNs work well in metal AM <ref:2610.09126#pg6>.

Jane: It’s about moving away from those manually crafted, static sampling strategies and toward a framework that learns where to place those points based on the process conditions, using conditional Flow Matching models and domain knowledge <ref:2610.09126#pg3>.

Lu: The theoretical foundation they provide shows that the adaptive proposal is strictly superior to conventional proposals because it’s proportional to the squared residual under each process condition and instant <ref:2610.09126#pg4>.

Meng: From a deployment view, this means we’re looking at a more robust training pipeline where the sampling itself becomes smarter about what needs attention <ref:2610.09126#pg4>.

Lalam: The main contribution is providing that theoretical foundation for collocation sampling from a generalization standpoint, along with a practical framework that scales across process conditions via amortized generative modeling and domain knowledge integration <ref:2610.09126#pg6>.

Tom: It’s about giving PINNs the ability to truly generalize across the diverse process conditions inherent in metal AM, which is what they claim is essential for understanding the entire process-structure-property chain <ref:2610.09126#pg1>.

Conclusion: Tom: So, we’ve gone through how this paper uses adaptive sampling to make physics models work better for metal additive manufacturing <ref:2610.09126#pg4>.

Jane: Exactly. The title itself is pretty technical, but the core idea is that you can’t use a one-size-fits-all sampling strategy when you are trying to model something complex like metal AM where the process conditions change so much <ref:2610.09126#pg4>.

Lu: The authors are using conditional Flow Matching models to generate these points adaptively, which basically means the AI learns where it needs more data based on what’s happening in the model itself <ref:2610.09126#pg4>.

Meng: From a deployment standpoint, I’m looking at this because if you can make a PINN generalize across different material properties without needing to retrain everything from scratch for every new process, that cuts down on the cost of getting a model ready <ref:2610.09126#pg4>.

Lalam: I’m pretty sure this paper is showing that when you tune the sampling based on the residual, which is how much the model is failing in specific spots, you get way better results than just throwing a million points at it randomly <ref:2610.09126#pg4>.

Tom: It sounds like they’re claiming this adaptive approach doesn't just give you a slightly better fit, but it actually helps the model generalize across different process conditions <ref:2610.09126#pg4>.

Jane: That’s right. The theoretical analysis they did proves that their adaptive proposal is strictly better than the standard static sampling methods we usually see in these simulations <ref:2610.09126#pg4>.

Lu: Specifically, Theorem two point one shows that conventional static sampling is strictly inferior to the adaptive sampling method because it’s mathematically tied to reducing statistical error rather than just increasing the total number of samples <ref:2610.09126#pg4>.

Meng: So what does that actually mean for a production system? It means we can use fewer collocation points and still get high accuracy, which is pretty good for efficiency <ref:2610.09126#pg4>.

Lalam: I think the main thing they’re pointing out is that the optimal sampling proposal should be proportional to the squared residual under each process condition and instant <ref:2610.09126#pg4>.

Tom: That’s a big shift. It moves us from guessing where to sample to letting the AI tell us exactly where it needs attention based on its own error signals <ref:2610.09126#pg4>.

Jane: And they did show this in experiments, achieving about a sixty-two percent reduction in relative L two error compared to other state-of-the-art parametric PINNs when using the same number of samples <ref:2610.09126#pg4>.

Lu: The ablation study is also important because it confirms that you need that domain knowledge from the base distribution to get both peak accuracy and stable training <ref:2610.09126#pg4>.

Meng: I still have to think about how much data the Conditional Flow Matching generator itself needs to be trained on before we can even use this thing in a real scenario <ref:2610.09126#pg4>.

Hyeonsu Lee, Jihoon Jeong

Texas A&M University

cs.LG

Submitted: 2026-10-06

Updated: 2026-10-06

Code: https://github.com/hsleecri/AdaptiveSamplingPINN-AM.1https:

License: http://creativecommons.org/licenses/by-nc-sa/4.0/

Importance score: 90/100

The gist: The gist: This work presents, to the best of our knowledge, the first adaptive collocation sampling strategy for PINNs in metal AM.

Key concepts

Adaptive Collocation Sampling: This is a strategy for choosing the specific locations in a neural network's training data (collocation points). Instead of picking points randomly or statically, this method intelligently selects points based on the current process conditions, such as temperature profiles during metal printing. The goal is to place more 'attention' where the physics are hardest to model.
Conditional Flow Matching (CFM): CFM is a generative modeling technique used here to approximate the complex distribution of high-residual errors that occur under different process conditions. It allows the system to efficiently generate relevant collocation points in a single step, making it possible to adapt sampling dynamically during training without needing separate models for every condition.
Domain-Informed Mixture Proposal: This is a method used in the second stage of sampling to balance exploration (trying new areas) and exploitation (focusing on known difficult areas). It combines points generated by a learned proposal with points from a base distribution, controlled by a mixture weight. This ensures the sampling process efficiently covers both general and specific thermal challenges.
Process Condition-Aware Sampling: This refers to tailoring the placement of training data based on the specific parameters of the metal AM process (e.g., laser power, scan speed). Conventional methods use fixed patterns, but this approach makes sampling decisions dependent on these conditions. The paper proves that making sampling condition-aware is strictly better for generalization than using fixed, static sampling points.

Terminology

Adaptive Sampling — 1× in this paper · explained in 1 episode
Flow Matching — 1× in this paper · explained in 15 episodes

Summary

The gist: This work presents, to the best of our knowledge, the first adaptive collocation sampling strategy for PINNs in metal AM.

Problem and Motivation

Accurate thermal modeling is essential in metal additive manufacturing (AM) for understanding the process-structure-property chain<ref:2610.09126#pg2>. Prior works typically rely on manually-crafted, static collocation sampling strategies, which are neither principled nor scalable across process conditions, hindering their generalization capability<ref:2610.09126#pg4>. This paper provides a theoretical analysis showing that process condition-aware adaptive sampling is strictly more favorable than conventional static sampling for generalization<ref:2610.09126#pg4>. Specifically, previous approaches statically concentrate collocation points in regions where steep and transient thermal gradients are expected (e.g., near the laser path), while remaining relatively coarse elsewhere<ref:2610.09126#pg4>. This conventional static collocation sampling strategy can be suboptimal and ultimately inhibits parametric PINNs from effectively generalizing across diverse process conditions<ref:2610.09126#pg4>.

Proposed Method

The proposed method introduces a two-stage framework for adaptive sampling within a conditional Flow Matching model<ref:2610.09126#pg4>. The first stage trains a Conditional Flow Matching (CFM) generator to approximate the process condition-dependent high-residual distribution and enables efficient generation of collocation points in a single pass<ref:2610.09126#pg4>. The second stage trains the final predictor using a domain-informed mixture proposal that balances exploration and exploitation<ref:2610.09126#pg4>. The overall training procedure is summarized in Algorithm 1<ref:2610.09126#pg9>. In Stage 2, collocation points are drawn from both the learned proposal gϕ and the base b with the proposed domain-informed mixture proposal qbα<ref:2610.09126#pg4>. The mixture weight α controls the exploration–exploitation trade-off, where smaller α more aggressively exploits learned high-residual regions, while larger α retains broader coverage<ref:2610.09126#pg4>.

Theoretical Analysis

The theoretical analysis shows that the excess generalization error decomposes into an optimization error and a statistical error<ref:2610.09126#pg5>. The statistical error is heavily governed by the variance of the loss estimator LbN (θ; q)<ref:2610.09126#pg5>. The paper proves that the adaptive proposal qθ(· t, µ) is strictly superior to conventional µ-independent proposals q(· t)<ref:2610.09126#pg4>. Specifically, Theorem 2.1 states that conventional static sampling is strictly inferior to adaptive sampling<ref:2610.09126#pg4>. The optimal adaptive proposal is shown to be proportional to the squared residual under each process condition and instant<ref:2610.09126#pg4>. This implies that reducing the variance of the empirical loss estimator σ¯2q is a more effective lever for minimizing the statistical error than merely increasing the total sample size N<ref:2610.09126#pg6>.

Experimental Validation

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 L2 error under an identical collocation budget<ref:2610.09126#pg4>. The proposed method reduces the relative L2 error of the current state-of-the-art parametric PINN baseline by up to 70.1%, purely by reallocating where collocation points are sampled<ref:2610.09126#pg4>. Qualitative analysis shows that high-residual regions emerge in heat dissipation regions that evolve with each material’s thermophysical properties<ref:2610.09126#pg4>. The ablation study confirms that incorporating domain knowledge via the base distribution is essential to simultaneously achieve peak accuracy and guaranteed training stability<ref:2610.09126#pg4>.

Conclusion

Taken together, our theoretical and empirical results establish that adaptive, process condition-aware collocation sampling is critical to realizing the generalization potential of parametric PINNs in metal AM<ref:2610.09126#pg6>. The work contributes a theoretical foundation for collocation sampling in PINNs from a generalization standpoint, a practical adaptive sampling framework that scales across process conditions via amortized generative modeling, and an integration of domain knowledge that ensures stable and efficient training<ref:2610.09126#pg6>.

Improvements for AI systems

Bold header: Adaptive Collocation Sampling via Flow Matching (CFM) The system can generate adaptive collocation points that are conditioned on both time steps and PDE parameters, effectively learning approximate high-residual distributions across different process conditions.
Bold header: Generalization Across Process Conditions The improved PINN can generalize across unseen process conditions by using a proposal density conditioned on both time and parameters, moving beyond the limitation of static sampling which is independent of µ.
Bold header: Superior Error Reduction in Metal AM Thermal Modeling The method achieves significant performance gains, reducing the relative L2 error of state-of-the-art parametric PINNs by an average of 62.1% and up to 70.1% on metal AM benchmarks under an identical collocation budget.
Bold header: Material-Dependent Thermal Behavior Capture The system can accurately capture distinct, material-dependent thermal behaviors, specifically by reallocating points toward heat dissipation regions that shift based on material properties like thermal conductivity and diffusivity.
Bold header: Enhanced Training Stability with Domain Knowledge The integration of a domain-informed base distribution ensures training stability, as this base distribution is shown to completely eliminate these bimodal failure modes as in previous research, complementing the learned proposal.

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.