Physics-Informed Machine Learning

• Physics-Informed Machine Learning (PIML) is a paradigm that integrates governing physical laws into neural networks to enforce physical consistency and reduce data dependency.
• It achieves accurate, mesh-free predictions by embedding PDEs, initial, and boundary conditions directly into the learning process, ideal for smart additive manufacturing.
• The method outperforms purely data-driven models and traditional FEM approaches by delivering 2–4% errors with significant speedups in computational time.

Physics-Informed Machine Learning (PIML) integrates mechanistic physical laws within machine learning models, leveraging the expressivity of neural networks while enforcing strict physical consistency and interpretability. This paradigm circumvents the need for large labeled datasets by embedding governing equations and boundary/initial conditions directly into the model’s structure and loss function. In the context of smart additive manufacturing, PIML enables mesh-free, data-efficient, and physically consistent predictions across high-dimensional, transient, and nonlinear domains, with quantifiable accuracy and interpretability advantages over both purely data-driven surrogates and conventional numerical solvers (Sharma et al., 15 Jul 2024). The approach has substantial implications for digital twins, process control, and generalization across varying regimes.

1. Mathematical Formulation and Core Architecture

The PIML framework is centered around a deep, fully-connected feed-forward neural network (“dense” MLP), in which all hidden layers use the Swish activation function $a(z)=z\cdot\sigma(z)$, with $\sigma(z)$ the standard sigmoid. The network receives $(x,y,z,t)$—spatiotemporal collocation points—as inputs and predicts the temperature field: $u_\theta(x,y,z,t) = T(x,y,z,t)\,,$ where $\theta$ collects trainable weights and biases.

The physical structure is encoded through the transient heat equation (energy conservation),

$$\rho\,C_p\,\frac{\partial T}{\partial t} = k\,\nabla^2T \quad\text{in } \Omega$$

with density $\rho$, heat capacity $C_p$, and thermal conductivity $k$. Boundary conditions (BC) represent a Gaussian laser heat source, convective, and radiative losses: $$-\,k\,\frac{\partial T}{\partial n} = Q_{\rm laser} + Q_{\rm conv} + Q_{\rm rad}$$ where

$$\begin{align*} Q_{\rm laser} & = \frac{2\eta P}{\pi r_b^2}\,\exp\Bigl(-\frac{2 d^2}{r_b^2}\Bigr),\ Q_{\rm conv} & = h\,(T-T_0), \ Q_{\rm rad} & = \varepsilon\,\sigma\,(T^4-T_0^4). \end{align*}$$

with $\eta$ (absorptivity), $P$ (laser power), $r_b$ (beam radius), $d$ (distance from beam), $h$ (convective coefficient), $\varepsilon$ (emissivity), $\sigma$ (Stefan–Boltzmann constant), and $T_0$ (ambient temperature).

The forward map is summarized as: $$u_\theta:\; (x,y,z,t) \longmapsto T_\theta(x,y,z,t)\,.$$ Layer-wise propagation for the network is given by: $$z^{(n)} = a\bigl(W^{(n)}z^{(n-1)} + b^{(n)}\bigr),\quad z^{(0)} = [x, y, z, t]^T.$$

2. Physics-Constrained Loss Construction and Training

The composite loss functional is: $$\mathcal{L}_{\rm total} = w_{\rm PDE}\,\mathcal{L}_{\rm PDE} + w_{\rm IC}\,\mathcal{L}_{\rm IC} + w_{\rm BC}\,\mathcal{L}_{\rm BC}$$ with weights $$\{w_{\rm PDE},w_{\rm IC},w_{\rm BC}\}=\{1,10^{-4},1\}$$. The terms are:

• PDE residual:

$$\mathcal{L}_{\rm PDE} = \frac{1}{N_{\rm coll}} \sum_{i} \left|\rho\,C_{p}\,\partial_{t}T_\theta - k\,\nabla^{2}T_\theta\right|^2$$

• Initial condition loss:

$$\mathcal{L}_{\rm IC} = \frac{1}{N_{\rm IC}} \sum_{j} \big|T_\theta(x_j,y_j,z_j,0) - T_0\big|^2$$

• Boundary condition loss:

$$\mathcal{L}_{\rm BC} = \frac{1}{N_{\rm BC}} \sum_{k} \left[-k\,\partial_n T_\theta - (Q_{\rm laser} + Q_{\rm conv} + Q_{\rm rad})\right]^2_k$$

Collocation points are sampled mesh-free with higher density near regions of large thermal gradient. Automatic differentiation is used to obtain all higher-order spatial and temporal derivatives.

The optimization is staged:

• Weight initialization by Glorot uniform.
• Adam optimizer (learning rate $2\times10^{-4}$) for $\sim$6,000 iterations.
• L-BFGS optimizer for $\sim$29,000 iterations.
• Total training time for the 3D case is 1.26 hours on a NVIDIA RTX A6000.

No labeled temperature data are used; learning is purely physics-driven.

3. Practical Implementation: Laser Metal Deposition Case Study

The case paper targets single-layer deposition of Ti–6Al–4V powders, with the computational domain of $25$ mm $\times$ $6$ mm $\times$ $4$ mm (using a half-model by symmetry), power $P=1000$ W, $\eta=0.4$, $r_b=1.5$ mm, and scan speed $5$ mm/s. Model validation is performed against a COMSOL (FEM) solution with $8,778$ mesh elements.

Key numerical metrics:

• Full-domain maximum absolute error: $61.2$ K (3.7% relative).
• Near melt-pool region (top surface) MAE: $34.7$ K (2.1% relative).
• Predicted temperature time-profiles at the surface and sub-surface probes match FEM reference within a few percent.
• Total loss converges to $\sim10^{-2}$, with well-balanced PDE and BC residuals.

This architecture achieved temperature prediction across all spatiotemporal points of the domain without any direct temperature observations—an indicator of strong data efficiency and regularization by physics.

4. Comparison with Pure ML and Conventional Solvers

The PIML model contrasts with other approaches:

Method	Labeled Data Requirement	Speed	Physical Consistency	Accuracy
Physics-driven PIML	None	$\sim$100$\times$ faster than FEM	By construction (PDE, BC, IC enforced)	2–4% error
Pure Data-driven ML	Thousands of labeled examples	High (inference)	Not guaranteed	Highly variable (depends on data)
High-fidelity FEM/CFD	None	Slow	By construction	Reference

PIML provides transparent, physically consistent predictions and is suitable for deployment in real-time or digital twin settings. The embedded physics constraints eliminate reliance on experimental or simulation-labeled datasets, which constitute a bottleneck in supervised ML.

5. Generalization, Adaptivity, and Extension

The framework is generalizable:

• The neural architecture is reusable; only the governing equations, parameters, or boundary conditions need revision to adapt to new process parameters or materials.
• The mesh-free approach enables transfer to domains with complex geometries and adaptive sample density, especially valuable near critical features (melt-pool, interfaces).
• The scheme can be extended to include multi-physics (e.g., momentum conservation, phase change, stress) by the inclusion of additional PDE residuals in the loss.

Transfer learning can be exploited: once the model is trained for a given process parameter set, quick adaptation for new operation regimes is possible, supporting real-time digital twin objectives or parametric studies.

Limitations:

• Some degradation of accuracy in the domain interior (away from boundaries and sampled collocation points). Adaptive collocation strategies or boundary-focused sampling are promising areas for future research.
• Adding multi-physics capability (e.g., fluid flow, phase change) requires the explicit formulation and embedding of additional associated PDEs and their boundary/initial conditions.
• Re-training is required when physical process parameters change, although transfer learning may accelerate this.

6. Significance and Future Outlook

PIML, as realized in this application, establishes a paradigm for mesh-free, data-efficient digital twins in additive manufacturing processes. Embedding the physical laws at the loss level confers major advantages:

• Dramatically reduced dataset requirements (sometimes to zero labeled data).
• Improved interpretability and physical plausibility.
• Generalizability across varying regimes by modular PDE specification.
• Orders of magnitude speed-up over full-field finite element simulations.

Future research will explore adaptive collocation, boundary-focused sampling, multi-physics extensions, and automated transfer learning. These directions are essential for next-generation digital twins, real-time process monitoring, and robustly interpretable AI-driven manufacturing (Sharma et al., 15 Jul 2024).