Physics Informed Machine Learning through Inductive Bias

Olga Fink

1 and Vinay Sharma¹

1 Intelligent Maintenance and Operations Systems, EPFL, Lausanne, Switzerland E-mail: olga.fink@epfl.ch

Status

In smart manufacturing, Physics-Informed Machine Learning (PIML) [1] is emerging as a critical enabler for building AI systems that are not only data-efficient but also physically consistent, interpretable, and trustworthy. Manufacturing environments are characterized by complex processes, interconnected systems, and often sparse or noisy sensor data [2] PIML addresses these challenges by embedding domain knowledge—such as governing equations, structural constraints, or physical symmetries—directly into the learning process.

PIML incorporates three types of bias to guide models toward generalizable and physically plausible solutions: inductive, observational, and learning biases. Inductive biases are prior assumptions embedded in the model architecture—such as spatial locality [3], conservation laws [4], or temporal continuity [5]—that constrain the solution space. Observational biases stem from how data is sampled or represented, while learning biases arise from the optimization strategy, such as regularization or specific loss functions.

In smart manufacturing, inductive bias plays a particularly important role due to the complexity of physical processes and the presence of diverse sensor configurations. Knowledge about sensor location (relational), temporal dynamics, and signal characteristics can be effectively incorporated as inductive biases to enhance model generalization, interpretability, and data efficiency [6]. It enables learning from limited data while ensuring model predictions respect known physical and operational constraints. For example, models that embed knowledge of thermal dynamics, material behavior, wear processes, or conservation laws are better suited for predictive maintenance, quality assurance, and control [7][8]. Inductive biases also enhance interpretability—critical for operator trust and deployment in safety-critical settings.

A key subclass of inductive bias includes structural and relational biases [3], which are increasingly important for modeling modern manufacturing systems, where sensor measurements often exhibit strong spatial and temporal correlations [9]. Structural bias encodes assumptions about the hierarchical or modular organization of systems, while relational bias captures interactions and interdependencies among components. These are naturally leveraged by Graph Neural Networks (GNNs), which show strong potential in smart manufacturing.

GNNs can model machines, sensors, and subsystems as nodes in a graph, with edges representing physical, functional, or spatial relationships. This makes them well-suited for tasks such as fault propagation analysis,

root-cause diagnosis, and system-level health monitoring. Spatiotemporal GNNs [10] further incorporate time dynamics, enabling real-time analysis of evolving sensor data across distributed systems.

In smart manufacturing, GNNs are increasingly used as surrogate models to approximate the behavior of complex dynamical systems where direct simulations (e.g., finite element or multi-physics models) are computationally expensive [11][12][13]. By learning from system-level sensor data and known component interactions, they emulate physical processes with high fidelity, enabling fast, scalable diagnostics, control, and optimization.

As smart manufacturing evolves toward connected, cyber-physical environments, integrating structural and relational biases—particularly through GNN-based PIML models—will be central to building robust, scalable, and transparent AI-driven decision-making systems.

Current and future challenges

Surrogate modeling is a key enabler in smart manufacturing, offering efficient approximations of complex physical and cyber-physical systems. These models are critical for real-time applications such as system diagnostics, control, and predictive maintenance, where full-scale simulations (e.g., finite element or multi-physics models) are computationally prohibitive. Despite their promise, surrogate models face a number of open challenges specific to manufacturing environments.

A primary requirement is fast, online dynamics prediction under changing operating conditions. Models must deliver reliable, real-time outputs even as system loads, speeds, or thermal states vary. Additionally, they must support long trajectory roll-outs —that is, iteratively predicting the system's future state over extended time horizons by feeding model outputs back as inputs—while minimizing error accumulation to ensure predictive stability over time.

Interpretability and explainability of learned dynamics remain critical. Many high-performing models behave as black boxes, making it difficult to verify or explain their outputs—an unacceptable limitation in safety-critical manufacturing contexts. Surrogates must not only be accurate but also transparent and physically meaningful.

Industrial systems are inherently noisy and data-rich. Models must handle noisy, heterogeneous sensor inputs, often with varying sampling rates and resolutions. They must also be capable of learning directly from observed trajectories, even when system parameters or governing equations are unknown, requiring strong inductive and relational biases.

Beyond robustness and expressivity, scalability is essential. Manufacturing processes increasingly involve multi-physics interactions (e.g., thermal-mechanical coupling) and multi-fidelity data streams—from high-resolution simulations to low-quality real-time sensors. Surrogates must integrate such information coherently and scale across spatial and temporal resolutions.

One of the most demanding challenges is generalization and extrapolation. Surrogate models are not just expected to generalize within the domain of their training data, but also to extrapolate to entirely new system configurations and operating conditions without retraining. This is particularly important in flexible or modular manufacturing settings, where system layouts and use-cases evolve continuously. Many current models lack the adaptability to handle such deployment scenarios, especially when failure data is sparse or evolving [14].

Additionally, inverse parameter inference—such as identifying process conditions, material properties, or system-level parameters from observed data—and the explicit modeling of degradation, wear, or other long-term system evolutions remain significant challenges. These limitations restrict the broader applicability of surrogate models in smart manufacturing, where accurate, interpretable, and dynamic models are essential for process optimization, adaptive control, quality assurance, and lifecycle management. Error! Reference source not found. illustrates the desired capabilities of surrogate models.

Figure 1. Desired Capabilities of Surrogate Models for PIML-Enabled Smart Manufacturing: Models must deliver

interpretable and stable outputs, handle noisy and incomplete inputs, and generalize to unseen configurations and operating conditions*.*

Advances in science and technology to meet challenges

Physics-Informed Graph Neural Networks (PI-GNNs) offer a promising approach for addressing the challenges outlined above. Recent developments integrating geometric and physical inductive biases— ranging from domain-specific physics priors, such as Kirchhoff’s laws [15] or heat-flux continuity [16], to broader principles like symmetry [17], thermodynamic laws [18], and momentum balance [4]—extend traditional GNNs into PI-GNNs. These physical inductive biases augment GNN’s inherent relational biases, enabling physically grounded and computationally efficient modeling of inter-component interactions, leading to following key capabilities:

• Data efficiency and long rollout prediction: Physics-aware inductive bias constrains the hypothesis space, enabling accurate learning from sparse, noisy data and preventing unphysical drift in long -term predictions.
• Generalization and Extrapolation: Modular interaction learning and embedded physics enhance transfer to unseen topologies, boundaries, and operating regimes.
• Interpretability and Explainability: Physical laws guide message passing, enabling pairwise interactions to represent meaningful internal variables (e.g., stress or heat flux), thus supporting transparency in safety-critical systems. To fully realize the potential of PI-GNNs as production-grade tools, research must advance in the following three key areas:

1. Cross-Domain application with universal physics-informed priors: While some recent PI-GNNs—such as thermodynamics-consistent networks [18] and momentum-preserving equivariant graph nets [4]—show cross-domain potential, most remain constrained by domain-specific biases. To be applicable across coupled thermo-mechanical, electro-mechanical, and fluid-structure systems, future architectures must embed universal physical laws—conservation of energy, momentum, mass, and charge—directly into their message-passing mechanisms.

2. Virtual sensing: Estimating unobserved variables—such as residual stresses in metal additive manufacturing or internal shear forces in high-viscosity mixers—is essential for process monitoring. Since these quantities are not directly measurable, PI-GNNs with strong physical priors can infer these internal variables from heterogeneous, multi-fidelity data. Well suited to this task, their learned pairwise messages implicitly represent quantities like contact loads [5]. Guided by physical laws (e.g., momentum conservation), these messages can be decoded into interpretable variables from observable dynamics, supporting adaptive control and defect prevention.

3. Community benchmarks: Advancing PI-GNN research in smart manufacturing requires benchmark datasets that combine high-fidelity simulations with real-world sensor data across diverse operating conditions. These datasets should capture domain shifts, sensor noise, and process variability to enable rigorous evaluation of generalization and robustness. Internal state measurements—such as contact loads from specialized test rigs—are especially important for advancing virtual sensing. When integrated, these key advances will lay the foundation for robust, interpretable, and scalable PI-GNN models that function as reliable digital surrogates—applicable across a wide range of system types— to support smarter and more sustainable system design, real-time monitoring, and adaptive control in next-generation smart manufacturing environments.

Concluding remarks

Physics-Informed Machine Learning is emerging as a key enabler for building AI systems that are data-efficient, interpretable, and aligned with physical principles—an essential requirement in smart manufacturing environments characterized by complex dynamics, sparse sensing, and safety-critical constraints.

While Graph Neural Networks are not inherently physics-informed, they offer a natural framework for representing the structured, relational nature of manufacturing systems. When extended with additional general physics-based inductive bias—such as conservation laws, symmetry, or energy consistency—GNNs can serve as effective surrogate models and reasoning engines across multi-physics, multi-scale environments.

These physics-informed GNNs show strong potential for supporting tasks such as fault propagation analysis, virtual sensing, long-horizon control, and adaptive monitoring. They also offer differentiable, structured models that can be integrated into modern control architectures.

To fully realize this potential, further research is needed in areas such as generalization, extrapolation, dynamic graph adaptation, and sim-to-real transfer. Developing standardized benchmarks that combine simulation and sensor data will be critical.

As smart manufacturing advances toward autonomous, cyber-physical systems, physics-guided learning frameworks like PI-GNNs will play an increasingly important role in building robust, trustworthy AI solutions.

Acknowledgements

This work has been supported by the Swiss National Science Foundation (SNSF) Grant 200021_200461.

References

[1] Karniadakis G E, Kevrekidis I G, Lu L, Perdikaris P, Wang S and Yang L 2021 Physics-informed machine learning Nat. Rev. *Phys.*3422–440 (doi: 10.1038/s42254-021-00314-5)

[2] Phuyal S, Bista D and Bista R 2020 Challenges, opportunities and future directions of smart manufacturing: A state-of-the-art review *Sustain. Prod. Consump.*23287–302 (doi: 10.1016/j.spc.2020.06.002) [3] Battaglia P W, Hamrick J B, Bapst V, Sanchez-Gonzalez A, Zambaldi V, Malinowski M, Tacchetti A, Raposo D, Santoro A, Faulkner R et al 2018 Relational inductive biases, deep learning, and graph networks arXiv:1806.01261 [4] Sharma V and Fink O 2025 Dynami-CAL GraphNet: A physics-informed graph neural network conserving linear and angular momentum for dynamical systems arXiv:2501.07373 [5] Sharma V, Oddon R T, Tesini P, Ravesloot J, Taal C and Fink O 2025 Equi-Euler GraphNet: An equivariant, temporal-dynamics informed graph neural network for dual force and trajectory prediction in multi-body systems arXiv:2504.13768 [6] Zhu G, Wang S and Wang L 2024 Heterogeneous graph neural network for modeling intelligent manufacturing systems *Meas. Sci. Technol.*36015114 [7] Peng J-Z, Hua Y, Li Y-B, Chen Z-H, Wu W-T and Aubry N 2023 Physics-informed graph convolutional neural network for modeling fluid flow and heat convection Phys. Fluids 35085104 [8] Xue T, Gan Z, Liao S and Cao J 2022 Physics-embedded graph network for accelerating phase-field simulation of microstructure evolution in additive manufacturing *npj Comput. Mater.*8201 [9] Zhao M, Taal C, Baggerohr S and Fink O 2025 Graph neural networks for virtual sensing in complex systems: Addressing heterogeneous temporal dynamics *Mech. Syst. Signal Process.*230112544 [10] Yu B, Yin H and Zhu Z 2018 Spatio-temporal graph convolutional networks: A deep learning framework for traffic forecasting Proc. IJCAI 3634–3640 [11] Sanchez-Gonzalez A, Godwin J, Pfaff T, Ying R, Leskovec J and Battaglia P 2020 Learning to simulate complex physics with graph networks Proc. ICML 8459–8468 [12] Shivaditya M V, Alves J, Bugiotti F and Magoulès F 2022 Graph neural network-based surrogate models for finite element analysis Proc. DCABES 54–57 [13] Maurizi M, Gao C and Berto F 2022 Predicting stress, strain and deformation fields in materials and structures with graph neural networks *Sci. Rep.*1221834 [14] Zhao Y, Li H, Zhou H, Attar H R, Pfaff T and Li N 2024 A review of graph neural network applications in mechanics-related domains *Artif. Intell. Rev.*57315 [15] Donon B, Clément R, Donnot B, Marot A, Guyon I and Schoenauer M 2020 Neural networks for power flow: Graph neural solver *Electr. Power Syst. Res.*189106547 [16] Niresi K F, Bissig H, Baumann H and Fink O 2024 Physics-enhanced graph neural networks for soft sensing in industrial internet of things IEEE Internet Things J. [17] Satorras V G, Hoogeboom E and Welling M 2021 E(n) equivariant graph neural networks Proc. ICML 9323–9332 [18] Tierz A, Alfaro I, González D, Chinesta F and Cueto E 2025 Graph neural networks informed locally by thermodynamics *Eng. Appl. Artif. Intell.*144110108