Physics-Informed Neural Networks for Additive Manufacturing
Embedding physical laws into neural networks for AM simulation and process modeling
Research Papers
1,100+
Primary Focus
Thermal modeling
Key Processes
LPBF, DED, EBM
Physics Domains
Heat, stress, fluid
Growth Rate
+85% since 2022
Speedup vs FEM
100-10,000x
Physics-Informed Neural Networks (PINNs) represent a paradigm shift in computational modeling for additive manufacturing, embedding governing physical equations directly into neural network loss functions. Unlike purely data-driven approaches, PINNs enforce conservation laws (mass, momentum, energy) as soft constraints, enabling accurate predictions with limited training data while maintaining physical consistency.
In AM applications, PINNs solve the fundamental challenge of expensive high-fidelity simulations: traditional finite element analysis of a single LPBF build can take days to weeks, while trained PINNs provide predictions in milliseconds. This enables real-time digital twins, rapid parameter optimization, and physics-consistent uncertainty quantification that purely empirical models cannot achieve.
PINN Fundamentals
PINNs minimize a composite loss function combining data fidelity and physics residuals:
L = L_data + λ_pde · L_pde + λ_bc · L_bc + λ_ic · L_ic
Where L_pde encodes PDEs (heat equation, Navier-Stokes), L_bc enforces boundary conditions, and L_ic specifies initial conditions. The hyperparameters λ balance data-driven learning with physics enforcement.
Governing Equations in AM
- Heat equation: ρc_p ∂T/∂t = ∇·(k∇T) + Q - governs thermal evolution
- Navier-Stokes: Melt pool fluid dynamics and Marangoni convection
- Elasticity: σ_ij,j + f_i = 0 for residual stress development
- Phase field: Solidification microstructure evolution
Advantages over Pure ML
- Physical consistency: Solutions satisfy conservation laws
- Data efficiency: 10-100x fewer training samples needed
- Extrapolation: Better generalization beyond training domain
- Interpretability: Physics-based loss provides diagnostic information
Thermal Modeling
Thermal history prediction is the most developed PINN application in AM, solving the transient heat conduction equation with moving laser/electron beam sources:
| Application |
Physics Encoded |
Accuracy |
Speedup |
| Single track temperature |
3D heat equation + Goldak source |
R² > 0.98 |
1,000x vs FEM |
| Multi-layer thermal history |
Heat equation + layer activation |
MAPE < 3% |
5,000x |
| Cooling rate prediction |
∂T/∂t extraction from thermal field |
Error < 8% |
500x |
| Full build simulation |
Transient 3D + boundary conditions |
R² > 0.95 |
10,000x |
Adaptive Heat Source Modeling
PINNs can learn spatially-varying absorptivity and thermal conductivity by treating material properties as trainable parameters constrained by physics. This enables inverse problems: inferring process conditions from limited temperature measurements.
Melt Pool Physics
Melt pool dynamics require coupling thermal and fluid equations, representing one of the most challenging PINN applications:
Multi-Physics Coupling
- Thermal-fluid: Temperature-dependent viscosity, surface tension
- Marangoni convection: Surface tension gradients drive flow
- Recoil pressure: Vapor pressure effects on pool shape
- Free surface: Level-set or VOF for interface tracking
| Melt Pool Feature |
PINN Approach |
Key Challenge |
| Pool geometry (W, D, L) |
Thermal PINN + Stefan condition |
Sharp interface handling |
| Flow velocity field |
NS-PINN with temperature coupling |
High Marangoni number |
| Keyhole formation |
Multi-phase PINN |
Vapor-liquid interface |
| Spatter prediction |
Hybrid PINN + particle tracking |
Discrete-continuum coupling |
Residual Stress Prediction
Residual stress and distortion are critical quality concerns in metal AM. PINNs solve thermo-mechanical coupled problems orders of magnitude faster than FEM:
Thermo-Mechanical Coupling
Sequential or fully-coupled approaches: thermal history → thermal strain → elastic-plastic stress evolution. PINNs enforce equilibrium (∇·σ = 0), compatibility, and constitutive relations simultaneously.
| Stress Application |
Physics Constraints |
Validation Method |
| In-situ stress evolution |
Equilibrium + thermal strain |
XRD, neutron diffraction |
| Part distortion |
Elasticity + plasticity |
CMM, optical scanning |
| Stress relaxation |
Creep/relaxation laws |
Hole drilling, contour |
| Crack initiation risk |
Fracture mechanics criteria |
CT, acoustic emission |
Inherent Strain Method + PINN
Combining the computational efficiency of inherent strain with PINN physics-enforcement enables part-scale distortion prediction in minutes rather than days, critical for design iteration and build planning.
PINN Architectures
| Architecture |
Description |
AM Application |
Advantage |
| Vanilla PINN |
MLP with physics loss |
Single-track thermal |
Simple implementation |
| DeepONet |
Operator learning for PDEs |
Parametric thermal fields |
Instant parameter sweeps |
| Fourier Neural Operator |
Spectral convolution layers |
High-frequency features |
Better high-gradient capture |
| hp-VPINN |
Variational + domain decomp |
Multi-layer builds |
Handles discontinuities |
| Separable PINN |
Factorized space-time |
Long build histories |
Memory efficient |
| Physics-Informed PointNet |
Point cloud + physics |
Complex geometries |
Mesh-free, CAD-ready |
Training Strategies
- Curriculum learning: Start with simplified physics, gradually add complexity
- Adaptive sampling: Focus collocation points on high-residual regions
- Transfer learning: Pre-train on analytical solutions, fine-tune on AM data
- Multi-fidelity: Combine coarse FEM with fine experimental data
AM Applications
By AM Process
- LPBF: Most studied; thermal, melt pool, microstructure PINNs
- DED: Larger melt pools; thermal-fluid coupling critical
- EBM: Vacuum environment; electron beam physics
- WAAM: Large scale; residual stress focus
Digital Twin Integration
PINNs enable real-time digital twins by providing physics-consistent predictions at sensor update rates. Key components:
- State estimation from limited sensor data
- Forward prediction of thermal/stress evolution
- Anomaly detection via physics residual monitoring
- Adaptive control through inverse PINN optimization
Uncertainty Quantification
Bayesian PINNs and ensemble methods provide prediction uncertainties essential for certification. Physics constraints reduce epistemic uncertainty compared to purely data-driven models, critical for aerospace/medical AM qualification.
Key References
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear PDEs
Raissi, Perdikaris, Karniadakis | Journal of Computational Physics | 2019 | 10,000+ citations
Physics-informed machine learning for metal additive manufacturing: Prediction of temperature and melt pool dimensions
Zhu et al. | Additive Manufacturing | 2021 | 380+ citations
A physics-informed neural network for heat transfer problems in additive manufacturing
Roy, Wodo | Engineering with Computers | 2022 | 250+ citations
DeepONet-based surrogate model for fast prediction of thermal history in laser powder bed fusion
Haghighat et al. | Computer Methods in Applied Mechanics | 2023 | 120+ citations
Physics-informed neural networks for residual stress prediction in additive manufacturing
Chen et al. | MSEA | 2023 | 85+ citations