This work presents a novel hybrid methodology combining Finite Element Method thermal simulations and Physics-Informed Neural Networks for improved thermal property and resistance prediction in electronic systems. By directly encoding physical laws into the learning procedure, the methodology facilitates accurate real-time inference of internal thermal parameters based on sparse external measurements.
As electronic systems continue to shrink and increase in functional density, managing heat becomes a critical challenge, especially in embedded and power electronics. Conventional thermal calibration approaches, including standardized datasheet values or finite element simulations, often rely heavily on expert intuition and trial-and-error procedures, making them time-consuming and dependent on human experience rather than ofering precise, automated solutions. To address these limitations, this paper introduces a hybrid calibration approach using Physics-Informed Neural Networks, which embed thermal physics directly into the learning process. By automating simulation workflows through PyAnsys and training lightweight, explainable models suitable for deployment, the proposed method enables accurate real-time prediction of internal thermal properties from sparse external measurements—achieving a balance between physical fidelity and computational eficiency.
Despite the increasing academic interest in Physics-Informed Neural Networks [
This paper addresses these limitations by introducing a lightweight, explainable Physics-Informed Neural
Network architecture specifically engineered for integration into a real-time calibration loop. The proposed
architecture is trained using a novel, high-throughput Finite Element Method dataset, generated through
PyAnsys. In contrast to prior research, such as the work by Du and Lu [
A Metal-Oxide-Semiconductor Field-Efect Transistor (MOSFET) shown in [Figure 1] is a three-terminal device—gate, drain, and source—widely used in both digital and analog circuits due to its high eficiency, fast switching speed, and low power consumption. It functions by applying an electric field at the gate to control the current between the drain and source, with a thin silicon dioxide (SiO2) layer ensuring high input impedance. In the automotive industry, MOSFETs serve in various roles including energy control for hybrid and electric vehicles, motor control in electric power steering (EPS) and HVAC systems, battery management systems (BMS), switching in DC-DC converters and inverters, and power regulation for LED lighting and infotainment systems.
These devices are typically mounted on Printed Circuit Boards (PCBs) as illustrated in [Figure 2], which are composed of insulating materials such as FR4 fiberglass and have copper traces for electrical connectivity. PCBs provide both mechanical and electrical support for components and come in single-layer, double-layer, or multi-layer configurations, making them essential in modern electronics due to their reliability, compactness, and ease of mass production.
Thermal resistance, a critical parameter for MOSFET operation, is typically provided in datasheets as ℎ(−) , representing the junction-to-ambient thermal resistance. This value is influenced by PCB parameters such as material, copper thickness, and layout. Derived from Fourier’s Law and an analogous to Ohm’s Law, thermal resistance is expressed as: where is the material thickness, is the thermal conductivity, and is the heat conduction area. The specific thermal resistance from junction to ambient is given by: ℎ = ℎ(−) ℎ(−) = = −
− while the junction-to-board resistance is: Here, is the junction temperature, is the ambient temperature, is the board temperature, and is the power dissipated. By optimizing copper area, thermal via placement, and PCB design, engineers can significantly reduce ℎ(−) , improving heat dissipation and device reliability. (1) (2) (3)
Despite the mathematical similarity between thermal and electrical resistance, there are practical diferences. For example, thermal conductivity spans only about three orders of magnitude, while electrical conductivity varies by over twenty. Furthermore, complex phenomena such as spreading resistance and multidimensional heat flow make analytical solutions less accurate. Therefore, simulation tools like Finite Element Analysis (FEA) are often used to model thermal behavior and optimize designs.
Datasheets often present multiple ℎ(−) shown in [Figure 3] values depending on test setups—such as
JEDEC-standard footprint [
Though the mathematical analogy between thermal and electrical resistance is sound, practical diferences
render this analogy of limited utility. As noted by Philips researcher Clemens J. M. Lasance in a 2008 review
paper [
ANSYS [
Complementing ANSYS’s robust graphical interface, PyAnsys [
Our proposed approach, Physics-Informed Machine Learning (PIML) represents an essential paradigm
shift in scientific simulation, overcoming the shortcomings of standard machine learning (ML) algorithms
and classical numerical methods. PIML combines the predictive capabilities of machine learning with
the strict structure of physical laws, usually by placing governing equations such as partial diferential
equations (PDEs) directly within the learning procedure. This hybrid framework provides notable benefits
in scientific applications where data can be limited, noisy, or costly to obtain, facilitating data-eficient and
physically consistent models with the ability to generalize well beyond observed data. The rationale for
PIML is driven by the computational expense, uncertainty quantification challenges, and inability to
incorporate incomplete or noisy data of traditional numerical simulations [
At the heart of PIML are Physics-Informed Neural Networks, neural Networks that are trained to minimize
both observed data and the underlying physical laws represented as PDEs. This is done by introducing PDE
residuals in the loss function of the network along with a data-driven term. Physics-Informed Neural Networks
have multiple advantages, such as mesh-free implementation, strong handling of noisy and/or incomplete
data [
This section details the proposed fully integrated computational pipeline for estimating the thermal conductivity of materials within semiconductor packages using artificial intelligence. The pipeline couples automated high-fidelity simulations with a data-driven prediction model. Specifically, we target a sixlayer semiconductor structure, leveraging finite element thermal simulations to generate temperature responses under controlled thermal loading. These responses are then used to train a neural network capable of inferring thermal conductivity across individual layers. The overarching goal is to develop a surrogate model that predicts internal material properties from Finite Element Method thermal measurements, with potential application in digital twins and real-time diagnostics.
Thermal behavior of semiconductor package was simulated by ANSYS MAPDL, invoked through Python under the PyAnsys API for complete automation. For every simulation run, thermal conductivity of the six inner layers had been perturbed randomly with values picked from a uniformly bounded distribution in ±10% of typical engineering baselines. This was done to provide a realistic account of manufacturing tolerance along with environmental variations.
The end-to-end simulation process included: • Generating the 3D layered geometry. • Assigning randomized conductivities to each layer. • Applying thermal loads and boundary conditions.
• Meshing, solving the heat equation, and extracting temperature results.
This induced heterogeneity was critical for training a generalizable learning model.
Simulations imposed a constant 0°C boundary condition on the top and bottom surfaces of the structure, replicating the heat sinks. Internal volumetric heat generation replicated device-level power dissipation. The resulting steady-state heat transfer equation was solved numerically:
∇ · (k∇T) + Q = 0 Where is the space-dependent conductivity and the internal heat generation. The boundary-driven temperature gradients recorded the thermal response with sensitivity to the conductivity prescribed.
To meet dataset size requirements, a multithreaded Python orchestration layer coordinated concurrent MAPDL session launches with dynamic port assignments. This allowed for near-linear scaling with CPU core count. Over 50,000 simulations were run and recorded successfully, with temperature outcomes and conductivity vectors written to a structured CSV format.
The goal of the AI model is to regress the per-layer conductivity values using only high-level temperature metrics—specifically, the maximum observed temperatures near the junction and the board interface. These inputs serve as proxy measurements, from which the model must infer the internal material configuration.
The raw simulation output was filtered to ensure data quality. Malformed or duplicated rows were removed, and entries with negligible heat flow (Temperatures < 0.2°C) were discarded to avoid learning from physically unrealistic cases.
The final cleaned dataset was standardized using a StandardScaler to center and scale inputs for improved optimization performance. Output targets (conductivity values) were retained in raw form to preserve interpretability.
Prediction accuracy was assessed using both mean and maximum absolute percentage error (APE) per volume:
APE = ⃒⃒⃒⃒ ^ − ⃒⃒⃒⃒ × 100%
The model performs exceptionally well for Volumes 1, 4, 5, and 6, with mean errors near zero and maximum deviations under 2.5%, as shown in Table 1. In this context, each volume corresponds to one of the six internal layers of the semiconductor package, whose thermal conductivities were individually varied during simulation. These layers exhibit consistent and distinct temperature–conductivity relationships, allowing the network to learn smooth and reliable functional mappings from surface temperature data to internal material properties.
Volume 2, however, displays a marked degradation in accuracy. The root causes likely include: • Higher Sensitivity: Volume 2 may lie near thermal bottlenecks where small conductivity shifts produce amplified efects on junction temperatures. • Nonlinearity: The relation between conductivity and observed temperature might be highly nonlinear or multivariate for this layer. • Data Sparsity: Volume 2 may have been underrepresented in the random sampling space, especially in edge cases.
0.02 1.25 0.02 0.02 0.17
0.82 25.18 2.22 2.21 0.86
Despite this, the model’s average performance remains suitable for many practical applications. The consistent accuracy across the majority of layers confirms the feasibility of indirect conductivity estimation via thermal signatures. Improvements in Volume 2 accuracy could be obtained via data augmentation, targeted oversampling, or physics-informed loss regularization.
In summary, this paper introduces a computationally eficient hybrid framework for coupling Finite Element Method simulations with Physics-Informed Neural Networks to facilitate precise, real-time inference of internal thermal properties in electronic systems based on sparse external measurements alone. By closing the gap between high-fidelity physical modeling and embedded, practical deployment, this method is particularly well-suited to the demands of digital twin applications, predictive diagnostics, and automated thermal calibration workflows.
Future work can be directed towards extending the framework to a wider class of electronic packages, enhancing robustness of models with noisy or missing data, and investigating the use of transfer learning methods to mitigate the requirement of large simulation datasets. Also, the incorporation of real-time sensor feedback and edge AI deployment can be considered to advance the system’s applicability in industrial, automotive, and consumer electronics applications.
Declaration on Generative AI During the preparation of this work the authors used Generative AI tools in order to improve the clarity and quality of expression. After using these tools, the authors reviewed and edited the content as needed and take full responsibility for the content of the paper.