Source-linked AI summary
A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications
Navid Zobeiry, Keith D. Humfeld
TL;DR
High-fidelity finite-element tools are slow for these applications. This study develops a physics-informed machine-learning approach with heat-transfer-based features and adaptive loss normalization; engineered features support accurate predictions beyond the training zone.
Problem
High-fidelity FE tools are slow, limiting their applicability, while developing ML techniques suitable for small applications remains challenging.
Method
The approach uses a loss function satisfying the PDE, boundary conditions, and initial condition, heat-transfer-theory feature engineering, and adaptive normalization for training.
Results
Engineered features enable accurate heat-transfer predictions beyond the training zone, whereas both NN and PINN match FE results within it.
Takeaways & Limitations
The trained model supports fast evaluation of boundary conditions for active manufacturing control.
Takeaways & Limitations
Simultaneously implementing multiple extensions may push computational power and reduce model accuracy.
Abstract
from arXiv · showhide
A physics-informed neural network is developed to solve conductive heat transfer partial differential equation (PDE), along with convective heat transfer PDEs as boundary conditions (BCs), in manufacturing and engineering applications where parts are heated in ovens. Since convective coefficients are typically unknown, current analysis approaches based on trial and error finite element (FE) simulations are slow. The loss function is defined based on errors to satisfy PDE, BCs and initial condition. An adaptive normalizing scheme is developed to reduce loss terms simultaneously. In addition, theory of heat transfer is used for feature engineering. The predictions for 1D and 2D cases are validated by comparing with FE results. It is shown that using engineered features, heat transfer beyond the training zone can be predicted. Trained model allows for fast evaluation of a range of BCs to develop feedback loops, realizing Industry 4.0 concept of active manufacturing control based on sensor data.
1. Introduction
Manufacturing heat-transfer processes involve uncertain boundary conditions and spatially varying temperature histories, making conventional FE-based analysis difficult to use for fast control. The paper develops a PINN with heat-transfer-informed features to predict across boundary conditions and beyond its training zone.
- Process challenge: Unknown or variable convective heat-transfer coefficients in oven heating create hot and cold spots, thermal gradients, and thermal lag.Airflow variation changes the boundary conditions, while conduction produces different temperature histories across the part.
- Process challenge: High-fidelity FE models are slow and require trial-and-error simulations for uncertain boundary conditions, limiting near-real-time Industry 4.0 applications.The speed–fidelity trade-off also affects the accuracy of surrogate models trained on FE-generated data.
- Existing approaches: Theory-agnostic ML approaches face challenges with small physical datasets, while numerical-data surrogates trade fidelity and speed when generating training data.Theory-guided ML combines physical laws, numerical data, and physical data to address limitations of earlier approaches.
- Existing approaches: PINNs train directly on governing PDE errors, avoiding pre-generated FE training data and enabling faster evaluation when boundary conditions are inputs.This motivates their use for manufacturing applications requiring rapid analysis.
- Study objective: Engineered features enable accurate heat-transfer predictions beyond the training zone and support fast evaluation across convective boundary conditions.The stated application is near-real-time feedback control of process parameters and part temperature histories.
2. Method
The method trains a neural network to satisfy the heat-transfer PDE, convective boundary conditions, and initial condition simultaneously. It addresses imbalance among loss terms through adaptive normalization and extends the formulation beyond one-dimensional heat transfer.
- Heat-transfer formulation: The network prediction f(x, t, h1, h2) is trained to solve the one-dimensional heat equation for specified boundary conditions.The formulation uses thermal properties and convective boundary conditions to represent heat transfer in the part.
- Loss construction: PDE adherence is quantified by the residual between the temporal temperature derivative and the spatial conduction term.A perfectly trained solution has zero PDE error at every point.
- Loss construction: Convective boundary-condition errors evaluate whether predictions satisfy the boundary equations at the two boundaries for any heat-transfer coefficient.Perfect adherence makes these error terms zero at any time and coefficient.
- Loss construction: The initial condition is treated as a boundary condition on the time boundary t = 0 and receives its own adherence error.This incorporates the initial thermal state into PINN training.
- Loss construction: The cumulative loss averages squared PDE, boundary-condition, and initial-condition errors, with scaling factors λ used to normalize terms.Combining losses is necessary because unequal magnitudes or sensitivities can cause training to favor one term over others.
3. Implementation and Training of a Physics-Informed Neural Network
The implementation combines physics-informed neural-network architectures, engineered heat-transfer features, ELU activations, and adaptive loss normalization to solve heat-transfer PDEs. Validation against finite-element results shows accurate interpolation and improved extrapolation, while coefficient inputs enable rapid evaluation across boundary conditions.
- Architecture and engineered features: The PINN solves the heat-transfer PDE using physics-informed loss terms and engineered features based on the governing heat-transfer solution form.The feature-engineering architecture combines position-dependent sine terms with time-dependent exponential terms before subsequent network processing.
- Activation functions: ELU was selected because PINN training requires a nonzero second derivative, whereas higher-order ReLU derivatives are zero.With six hidden layers, ELU trained the PINN accurately; tanh trials with four to ten hidden layers were unsuccessful.
- Adaptive normalization: Adaptive normalization updates loss-term factors according to each term’s ratio to the greatest loss term, preventing large errors from dominating training.Factors are updated at regular intervals, with a threshold such as 0.01 determining whether a factor remains unity or is adjusted.
- Validation and extrapolation: Both models match finite-element predictions within the training range, but beyond 15 minutes the PINN remains closer to FE while NN diverges.The authors attribute the improved extrapolation to physics-informed features capturing the underlying heat-transfer behavior.
- Boundary-condition evaluation: A PINN that includes heat-transfer coefficients as inputs predicts temperature for coefficient combinations without generating separate training data, at very low computational cost.The extended network achieved similar accuracy to the two-input network and was validated against finite-element calculations across coefficient combinations.
5. Extension to 2D and Beyond
The PINN is extended from one to two dimensions by adding spatial inputs and corresponding PDE and boundary-condition terms, enabling temperature prediction at arbitrary queried points. Further extensions to material properties, temperature-profile families, and higher dimensions increase training demands and may exceed typical individual-computer capacity.
- 2D extension: The 2D PINN adds a y input and adjusts PDE and heat-transfer boundary-condition error calculations.Randomly generated training points mean the added input increases the epochs required rather than requiring a fixed expanded dataset.
- 2D extension: After training, temperature can be predicted at any queried x, y, t, h1, and h2.Figure 9 illustrates this capability with temperature distributions at 5, 10, and 15 minutes.
- 2D extension: The 2D predictions were validated by comparison with finite-element results.The illustrated case uses a 60 mm by 20 mm composite part and 100 W/m2K coefficients on the x = 0 and y = 0 boundaries.
- Beyond 2D: A 3D extension requires a z input, an additional pre-layer, a z second-derivative PDE term, and extra boundary-condition error terms.Additional heat-transfer boundary conditions may also require additional heat-transfer inputs.
- Beyond 2D: Material-property inputs can be added: 1D uses thermal diffusivity and conductivity, while 2D and 3D require volumetric heat capacity and conductivity components.Adding two or three inputs to the 2D or 3D network further increases the batch size and epochs needed for acceptable loss.
- Beyond 2D: The trained model is tied to the captured air-temperature profile, so each profile requires a newly trained model.A parameterized family of profiles could use five additional input variables, but simultaneous extensions may exceed most individual computers’ training capacity.
6. Conclusions
The study develops a physics-informed neural network for heat-transfer PDEs with convective boundary conditions, combining physics-informed features and adaptive normalization. Validated against finite-element results, the PINN predicts beyond its training zone and supports fast evaluation for active manufacturing control.
- The developed PINN solves heat-transfer PDEs with convective boundary conditions using a loss function satisfying the PDE, boundary conditions, and initial condition.The approach also uses heat-transfer theory to engineer neural-network features.
- Adaptive normalization addresses differences in loss-term magnitudes during training.Training-point selection also increases density around discontinuities of the loss function and inputs.
- The trained PINN was validated on several 1D and 2D heat-transfer cases by comparison with finite-element results.Performance and accuracy were also compared with a neural network without feature engineering.
- Both NN and PINN match finite-element results within the training zone, but only the PINN with engineered features accurately predicts beyond that zone.The conclusion attributes this extrapolation capability to capturing the physics through engineered features.
- Once trained, a PINN provides near real-time simulation for problems with given boundary conditions and quick evaluation during manufacturing.This capability supports feedback loops for active manufacturing control despite process uncertainties and unknown boundary conditions.