PINNs for Transient Thermal Problems

Detailed analysis of physics-informed neural networks for transient heat conduction with moving sources, and why they are not recommended for the TMM surrogate project.

PINN fundamentals for heat equations

PINNs embed the heat equation residual in the loss function:

L = L_data + lambda_1 * L_PDE + lambda_2 * L_IC + lambda_3 * L_BC

where L_PDE enforces:

rho * Cp * dT/dt - div(k * grad(T)) - Q = 0

Key papers for transient thermal

Moving heat source problems

Wang et al. (2024): PINN for transient heat conduction with moving laser heat source in AM. Demonstrated feasibility for single-track LPBF but required careful loss-term balancing and curriculum training.

Citation: Wang, K. et al. (2024). “Physics-informed neural networks for transient heat conduction with moving heat sources in additive manufacturing.” Int. J. Heat and Mass Transfer.

Transient heat equation with sharp gradients

Mehrdad et al. (2023): Addressed the challenge of sharp thermal gradients near the heat source using adaptive sampling and gradient- enhanced PINNs.

Citation: Mehrdad, S.H. et al. (2023). “Gradient-enhanced physics-informed neural networks for transient heat conduction with sharp gradients.” CMAME.

Multi-layer AM thermal cycles

No successful demonstrations of PINNs for multi-layer AM thermal cycles with repeated reheating. The long time horizon (>10^4 time steps) and cyclic nature cause severe error accumulation.

Why PINNs fail for coupled TMM

Three coupled PDE systems in one loss function

1. Heat equation (parabolic, moving source): smooth solutions except near heat source
2. Momentum balance (elliptic with plasticity): discontinuous gradients at yield surface
3. Kinetics ODEs (stiff, path-dependent): JMAK/KM equations have exponential nonlinearity and vastly different time scales

Training instability sources

• Scale mismatch: Temperature ~10^3 K, stress ~10^8 Pa, phase fraction ~[0,1]. Loss terms differ by 10+ orders of magnitude.
• Stiffness mismatch: Heat equation time scale ~10^-2 s, metallurgical kinetics ~10^-6 s near transformation temperatures.
• Convergence rate mismatch: Parabolic PDE (heat) converges faster than elliptic PDE (mechanics) which converges faster than stiff ODEs (kinetics).
• Loss balancing: No principled method for weighting three fundamentally different physics in one loss. Adaptive weighting schemes (Wang, Teng & Karniadakis 2021) help for 2-term losses but not 3+ coupled systems.

Path dependence problem

Metallurgical transformations are path-dependent: the current phase fractions depend on the entire thermal history, not just the current temperature. PINNs must either:

• Augment input space with time (4D spatio-temporal input), which increases the curse of dimensionality
• Use recurrent PINN variants, which are unproven at this complexity
• Include history variables as additional network outputs, which creates circular dependencies in the loss function

Conservative neural networks

Jin, Meng & Liu (2022): Conservative PINNs that enforce energy/mass conservation exactly through network architecture design rather than soft penalty terms.

Citation: Jin, M., Meng, X. & Liu, H. (2022). “Conservative physics-informed neural networks on discrete domains for conservation laws.” CMAME.

For the heat equation, energy conservation is:

d/dt integral(rho * Cp * T dV) = integral(Q dV) - integral(q dot n dA)

Conservative PINNs enforce this exactly at the discrete level, preventing energy drift over long time horizons. This is critical for AM simulations spanning hours.

Hard constraint enforcement vs penalty methods

Penalty methods (standard PINN)

Boundary and initial conditions enforced via loss terms:

L_IC = ||T(x, 0) - T_0(x)||^2
L_BC = ||T(x_b, t) - T_bc(x_b, t)||^2

Problems: soft enforcement means BCs/ICs are only approximately satisfied. Error accumulates over time, especially for long transients.

Hard constraint methods

Lu et al. (2021): Construct network output as:

T_hat(x,t) = g(x,t) + d(x,t) * NN(x,t)

where g satisfies BCs/ICs exactly and d(x,t) is a distance function that vanishes on boundaries. The NN only needs to learn the interior solution.

Citation: Lu, L. et al. (2021). “Physics-informed neural networks with hard constraints.” SIAM J. Scientific Computing.

Advantages: exact BC/IC satisfaction, faster training, better long-time behavior. Disadvantages: constructing g and d for complex AM geometries with evolving boundaries is extremely difficult.

Hybrid FEM-ML approach

The recommended alternative to pure PINNs: use FEM for conservation laws (heat equation, momentum balance) and ML for closure terms (constitutive laws, metallurgical kinetics).

Advantages:

• Conservation is guaranteed by the FEM discretization
• ML only needs to learn local relationships (phase fractions from thermal history), not global PDE solutions
• Training is simpler: supervised learning on FEM data
• No loss balancing between physics

This is exactly the architecture recommended in TMM Surrogates (Option 3: neural operator chain with local kinetics solver).

Research gaps

1. No successful PINN demonstration for full TMM coupling in AM
2. No PINN handling of evolving domain (layer deposition)
3. No long-horizon PINN with bounded error for AM thermal cycles
4. No principled loss balancing for 3+ coupled physics
5. Hard constraint methods not extended to complex AM geometries

Connection to TMM surrogate project

PINNs are not recommended for the TMM surrogate project. The neural operator chain (Option 3 in TMM Surrogates) is preferred because:

• Neural operators learn the mapping directly, no loss balancing across physics
• Path dependence handled naturally by learning full spatio-temporal operator
• Metallurgical step is local (ODE per material point), no PDE- constrained network needed
• Training is sequential and stable

See also Neural Operators for PDE Surrogates for the recommended architecture.