A Fast and Generalizable Fourier Neural Operator-Based Surrogate for Melt-Pool Prediction in Laser Processing

AI summary

Overview

Research area: Scientific machine learning — specifically neural operator learning (Fourier Neural Operator) applied to multiphysics simulation of laser welding and melting of metals.

Technical level: Advanced. The paper assumes familiarity with PDE operator learning, spectral methods, and thermo-fluid welding physics.

Scope in one sentence: The authors build and evaluate LP-FNO, a Fourier Neural Operator surrogate trained on FLOW-3D WELD® multiphysics simulations, that maps laser power and scan speed directly to three-dimensional temperature fields and melt-pool geometry for Ti-6Al-4V across conduction and keyhole welding regimes.

What This Paper Is About

High-fidelity thermo-fluid simulations of laser welding — which must capture phase change, free-surface deformation, Marangoni convection, recoil pressure, and keyhole dynamics — are accurate but far too slow for large parameter sweeps, optimization, or real-time control. The authors' goal is to replace the expensive simulation with a learned operator that, once trained, predicts the full three-dimensional temperature field and melt-pool boundary for any process parameters inside the training window in milliseconds. The central technical obstacle they address is that keyhole welding is inherently unsteady, which conflicts with the steady-state setting that standard operator learning assumes — a conflict they resolve by moving into the laser's reference frame and applying temporal averaging.

Key Contributions

1. The LP-FNO architecture and workflow — a Fourier Neural Operator surrogate that maps process parameters (laser power, scan speed, normalized enthalpy, grid coordinates) to three joint output fields: temperature, volume fraction (alpha), and liquid fraction, trained on FLOW-3D WELD® data.

2. A reformulation that makes keyhole welding amenable to operator learning — the transient problem is expressed in a reference frame moving with the laser, then temporally averaged over a sliding window of n = 30 time steps, producing a quasi-steady state that the FNO can learn. The authors state this is, to their knowledge, the first surrogate that spans both conduction and keyhole regimes while treating keyhole dynamics within a quasi-steady operator-learning framework via time averaging.

3. A generalizable sampling strategy based on normalized enthalpy — training, validation, and test points are sampled on a grid uniform in normalized enthalpy H* and power P (with V_scan derived from the H* definition), rather than on a naive power/speed grid, so that lack-of-fusion, conduction, and keyhole regimes are represented evenly.

4. Demonstration of resolution transfer (super-resolution) — a single model trained on coarse 10 µm data is evaluated on fine 5 µm grids, yielding accurate super-resolved predictions in mesh-converged conduction regimes, while keyhole-regime discrepancies are attributed to unresolved dynamics in the coarse-mesh training data.

Main Findings

• Temperature accuracy on the order of 1%: The abstract reports temperature prediction errors on the order of 1%. The paper presents relative MAE and relative RMSE metrics, but does not state a single headline percentage in the body text available here.

• Melt-pool segmentation IoU above 0.9: Intersection-over-union scores for melt-pool segmentation exceed 0.9, using a threshold of tau = 0.5 on both the alpha and liquid-fraction fields, with the melt-pool region defined by alpha >= 0.5 and f_l >= 0.5.

• Quasi-steadiness is confirmed by temporal-difference analysis: In the moving laser frame, the mean absolute temperature difference between successive snapshots persists with significant fluctuations in the keyhole regime (P = 150 W, V_scan = 0.542 m/s, H* = 7.54) and is substantially smaller in the conduction regime (P = 70 W, V_scan = 0.298 m/s, H* = 4.75). After averaging over a 30-step window, the magnitude of the difference approaches zero in both regimes. Quasi-static behavior appears around t = 30 for non-averaged data and from t_avg = 50 in averaged data.

• Roughly ten-thousand-fold to hundred-thousand-fold speedup: Table 3 reports a FLOW-3D® transient simulation (power 50 W, scan speed 0.260 m/s, scan distance 0.6 mm) taking 6 min 7 s on the coarse 10 µm mesh and 1 h 6 min 54 s on the fine 5 µm mesh. LP-FNO inference takes 0.0100 s at 10 µm and 0.0435 s at 5 µm. Training (6,000 steps) took 7 min 24 s.

• Inference in tens of milliseconds at multiple resolutions: For P = 150 W and V_scan = 0.54 m/s, the model produces fields in around 10 ms at 10 µm resolution and around 40 ms at 5 µm resolution.

• Errors vary systematically across the process window: Error maps over power/enthalpy and power/scan-speed space show average absolute temperature error (in Kelvin) varying across the regime indicators for lack of fusion, conduction, and unstable keyhole. Exact per-regime error values are not reported in the body text included here.

• Worst case is at a regime boundary: The largest deviation occurs at P = 50 W and V_scan = 0.122 m/s, an outlier case near the conduction-to-keyhole transition, where the model infers a spurious gas bubble. The overall interface geometry is still well captured, and the authors suggest a few additional training points in this region would likely reduce the error substantially.

• Resolution transfer is regime-dependent: Super-resolved predictions from coarse-mesh training are accurate in mesh-converged conduction regimes; discrepancies in keyhole regimes reflect dynamics not resolved in the coarse training data.

• Training stability: The model converged across all outputs in terms of raw loss and per-fold test set performance, evaluated with 8-fold cross validation (7/8 training, 1/8 evaluation per fold).

Methodology in Plain English

Step 1 — Generate ground truth. The team simulated single-track laser scans on Ti-6Al-4V using FLOW-3D WELD® 2025R1, a commercial Volume-of-Fluid solver that handles surface tension, evaporation recoil pressure, Marangoni convection, phase change, and ray-traced laser absorption. The domain is a 0.9 mm × 0.4 mm × 0.3 mm cuboid on a uniform 10 µm Cartesian mesh, with a 50 µm Gaussian laser beam moving lengthwise across 0.6 mm. Boundary conditions are 1 atm pressure on top and 300 K elsewhere. Results are written every 5 µs.

Step 2 — Choose which parameters to vary. Laser power P ranges from 40 W to 190 W and scan speed V_scan from 0.1 m/s to 1 m/s, chosen to cover lack-of-fusion, conduction, and keyhole modes. Points are sampled on a grid uniform in normalized enthalpy H* and power, where H* = eta·P / (rho·(Cp·Delta_Tm + L)·sqrt(pi·D·sigma^3·V_scan)), using absorptivity 0.35, density 4420 (as listed in the paper), specific heat capacity 750 J kg⁻¹ K⁻¹, temperature rise to melting 1573 K, thermal diffusivity 8.1e-6 m, beam radius 50 µm, and latent heat of fusion 3.45e5 J kg⁻¹. This ensures both keyhole and conduction regimes are represented evenly.

Step 3 — Make the problem steady. The coordinate system is placed on the moving laser, and the time step is chosen so the laser advances exactly one grid cell per step, which avoids any spatial interpolation. Then a sliding-window average over 30 time steps (Eq. 12) filters out fast interface oscillations, leaving a quasi-static target field.

Step 4 — Learn the operator. A standard FNO is used: an MLP lifts the inputs (power, scan speed, normalized enthalpy, grid coordinates) into a higher-dimensional channel space, several Fourier layers mix information globally by transforming to the frequency domain, keeping only low-frequency modes, multiplying by learned weights per mode, and transforming back, and a final MLP projects down to temperature, volume fraction, and liquid fraction. The nonlinearity is GELU, zero padding of size 9 handles the non-periodic top boundary, and the ReLoBraLo aggregator combines the multiple output-field losses (chosen because it was designed for PINNs). Fields are normalized by reference values (1e-4 m, 3000 K, 1e-1 m/s, 10 W, 5.9 for H) and the liquid fraction is computed from predicted temperature as an extra training signal to sharpen the melt-pool boundary. The implementation uses NVIDIA's PhysicsNeMo package and the LION optimizer.

Step 5 — Mask the gas. Because FLOW-3D WELD® does not model the gas explicitly and the quantities of interest lie in the metal, a smooth tanh gate (sharpness k = 20) based on the volume fraction blends the temperature toward the boiling point of Ti-6Al-4V, 3123 K, outside the metal, and masks the volume and liquid fractions to zero there. This is applied both to the ground truth and after inference. Solidus and liquidus temperatures used are 1873 K and 1923 K.

Step 6 — Evaluate. A held-out validation subset was used for hyperparameter tuning; the rest of the data was assessed with 8-fold cross validation. Metrics are MAE, RMSE, relative MAE/RMSE (with a global mean normalizer), and IoU at threshold 0.5. Relative metrics were not computed for alpha and liquid fraction because those fields are already dimensionless and mostly zero-valued.

Why This Matters

Impact on research. This is a demonstration that a single operator-learning model can cover a parameter range that includes the keyhole regime — the unsteady, vapor-depression regime that most prior PINN and neural-operator welding work explicitly excludes and confines to conduction. The moving-frame plus temporal-averaging trick is a general recipe for making other unsteady multiphysics processes tractable for operator learning, and the use of normalized enthalpy as the sampling coordinate is a transferable strategy for spreading training data across distinct physical regimes. The paper also touches on a topic rarely addressed in the surrogate literature: how errors behave when a coarse-trained model is evaluated on finer grids, and why keyhole-regime super-resolution fails.

Real-world applications (as implied by the paper):

• Real-time process control — the authors explicitly motivate the work by citing digital twins and real-time control, which the millisecond inference enables.
• Rapid process finding and optimization — building error maps over the power/scan-speed space becomes cheap enough to sweep exhaustively.
• Uncertainty quantification and multi-parameter studies — the paper notes that the computational cost of high-fidelity simulation is especially problematic when many parameter changes are involved.
• Additive manufacturing and laser powder bed fusion — the related-work discussion frames this as part of a broader push toward digital twins for L-PBF and directed energy deposition.

Industry relevance. Laser welding is a widely used manufacturing process, and the authors argue it offers advantages in joining dissimilar materials, delivering localized heat, operating at high speed, and working in geometrically constrained regions. Traditional FVM and FEM tools are accurate but slow with fine meshes, strong melt-pool gradients, deep keyholes, long tracks, and multiple passes. A surrogate that produces full three-dimensional fields for any parameter set in tens of milliseconds — versus minutes to hours for a single FVM run — removes a practical bottleneck for industrial process design.

Future Directions

• Adding training data at regime boundaries. The worst error (P = 50 W, V_scan = 0.122 m/s, near the conduction-to-keyhole transition, where a spurious gas bubble is predicted) is explicitly attributed to sparse coverage; the authors suggest a few extra points there would greatly reduce the error.
• Extending beyond a single material. Ti-6Al-4V was the only material simulated; the non-dimensional normalized-enthalpy parameterization is described as a first step toward model generalization "not limited to a single material," suggesting generalization across materials as an open goal.
• Improving super-resolution in the keyhole regime. Coarse-mesh training data does not resolve the dynamics needed for accurate fine-grid keyhole predictions; how to obtain accurate super-resolution there, or what mesh resolution is sufficient, remains unresolved.
• Incorporating experimental ground truth. The study deliberately relies exclusively on simulation data to avoid the cost and complexity of experiments, leaving open how well the surrogate would match measured melt pools.

Target Audience

Researchers and engineers working on surrogate modeling, operator learning, or scientific machine learning for manufacturing processes will find the most value, particularly those trying to push neural operators beyond the steady conduction regime into unsteady multiphysics. It is also relevant to welding and additive-manufacturing process engineers interested in digital twins and real-time control, and to computational scientists weighing when a learned surrogate can substitute for a Finite Volume solver. Readers need a working knowledge of neural operators and welding physics to follow the details, though the problem statement and results are accessible to a broader engineering audience.

Note on unavailable details: The provided paper text is truncated and omits the full hyperparameter table (Table 6, referenced in Section 5) and the remainder of the figures. Consequently, the exact number of simulation samples in the dataset, the numerical temperature MAE and IoU values broken down by regime, and the full training hyperparameters are not reported in the available content.

Authors’ abstract

High-fidelity simulations of laser welding capture complex thermo-fluid phenomena, including phase change, free-surface deformation, and keyhole dynamics, however their computational cost limits large-scale process exploration and real-time use. In this work we present the Laser Processing Fourier Neural Operator (LP-FNO), a Fourier Neural Operator (FNO) based surrogate model that learns the parametric solution operator of various laser processes from multiphysics simulations generated with FLOW-3D WELD (registered trademark). Through a novel approach of reformulating the transient problem in the moving laser frame and applying temporal averaging, the system results in a quasi-steady state setting suitable for operator learning, even in the keyhole welding regime. The proposed LP-FNO maps process parameters to three-dimensional temperature fields and melt-pool boundaries across a broad process window spanning conduction and keyhole regimes using the non-dimensional normalized enthalpy formulation. The model achieves temperature prediction errors on the order of 1% and intersection-over-union scores for melt-pool segmentation over 0.9. We demonstrate that a LP-FNO model trained on coarse-resolution data can be evaluated on finer grids, yielding accurate super-resolved predictions in mesh-converged conduction regimes, whereas discrepancies in keyhole regimes reflect unresolved dynamics in the coarse-mesh training data. These results indicate that the LP-FNO provides an efficient surrogate modeling framework for laser welding, enabling prediction of full three-dimensional fields and phase interfaces over wide parameter ranges in just tens of milliseconds, up to a hundred thousand times faster than traditional Finite Volume multi-physics software.