Physics-Informed Training Strategy

• Physics-informed training strategies are approaches that embed physical laws into neural networks to improve simulation fidelity and efficiency.
• They employ advanced loss functions and optimization techniques, such as ISGD and non-standard norms, to manage multi-scale and high-frequency challenges.
• Adaptive methods, including curriculum learning and dynamic grid refinement, enable robust integration of real-world experimental data and enhance model convergence.

Physics-informed training strategies have gained considerable attention due to their ability to synergize deep learning methods with physical principles, offering robust solutions in computational science, modeling, and inverse problem-solving. Central to these strategies is the concept of embedding the governing equations of the systems being modeled directly into the loss functions of neural networks. This approach ensures that the machine learning models adhere not only to data but also to the fundamental physical laws governing the processes. Various innovative techniques within this framework have emerged to address challenges like spectral bias, multi-scale complexity, and convergence issues that are prevalent in traditional approaches. Through these strategies, practitioners can more effectively handle complex simulations and real-world data integration, leading to improved prediction accuracy and computational efficiency.

Embedding Physical Laws into Neural Networks

Physics-informed neural networks (PINNs) are designed to integrate the governing equations of physical systems directly into their architecture. This involves introducing additional terms into the loss function that measure the deviation of the neural network’s predictions from the physical laws. For instance, the classic swing equation describes generator dynamics in power systems with differential equations, augmented by penalizing deviations from these equations through the loss function (Misyris et al., 2019). The approach leverages automatic differentiation to incorporate these physical constraints, significantly reducing the data requirement and simplifying the network architecture by effectively "informing" the network with physics.

Advanced Loss Functions and Optimization Techniques

The choice of loss functions plays a crucial role in the efficacy of physics-informed training strategies. While the conventional L² loss is widely used in PINN training, it can be inadequate for problems where precise stability and approximation control are critical, such as Hamilton-Jacobi-BeLLMan equations in optimal control. Alternative loss functions, like Lᵖ or L^∞ norms, have been proposed to enhance stability by focusing on worst-case residuals (Wang et al., 2022). Furthermore, optimization techniques such as implicit stochastic gradient descent (ISGD) have been developed to mitigate the stiffness in gradient flow dynamics, thereby allowing larger learning rates and ensuring more robust convergence in PINN training (Li et al., 2023).

Handling High-Frequency and Multi-Scale Problems

Training PINNs for high-frequency or multi-scale problems, such as those found in turbulent flows or materials with complex properties, can lead to accuracy and convergence challenges. Strategies like transfer learning, where models trained on low-frequency problems are adapted to handle high-frequency variations, have shown effectiveness in addressing these challenges without increasing network parameters (Mustajab et al., 5 Jan 2024). Likewise, multi-level datasets leveraging different densities and distributions of collocation points have been employed to expose PINNs progressively to multiple solution spectra, enhancing their ability to capture a broader range of features (Tsai et al., 30 Apr 2025).

Adaptive Learning Strategies

Adaptive training schemes, particularly those integrating curriculum training, provide a structured approach to managing complexity in PINN frameworks. By initially focusing on simpler subdomains and gradually increasing difficulty, networks can stabilize the learning process and achieve better convergence in geo-mechanical and poroelastic flow applications (Bekele, 22 Apr 2024). Furthermore, adaptive grid-dependent methods allow for dynamic refinement of the solution representation as the training progresses, facilitating efficient handling of multi-scale PDE problems (Rigas et al., 24 Jul 2024).

Incorporating Real-World Experimental Data

Integrating real-world experimental data poses specific challenges due to competing optimization objectives and the need for physical consistency. Novel strategies have been developed to progressively assimilate experimental losses, conditionally update material parameters based on temperature estimates, and employ custom learning rate schedulers to manage premature reductions effectively (Zak et al., 6 Aug 2025). These strategies enhance the ability of PINNs to reconstruct internal states in processes like aluminum spot welding, expanding their application in industrial settings for real-time quality control.

Leveraging Advances in Neural Network Architectures

Recent advances have explored the use of structured neural network architectures that incorporate domain-specific biases to amplify relevant features, such as symmetry breaking in magnetic phase systems. These architectures, combined with training configurations that explicitly break symmetries, enhance network sensitivity to phase changes, allowing for efficient characterization of complex system behaviors without explicit labels (Medina et al., 15 May 2025). Additionally, the integration of generative adversarial networks (GANs) with physics-informed transformers provides adaptive mechanisms to focus training on high-residual regions, improving robustness and temporal causality in simulations (Zhang et al., 15 Jul 2025).

In conclusion, physics-informed training strategies offer a powerful toolkit for bridging the gap between machine learning and scientific computing. By embedding physical principles into neural network architectures and employing advanced training techniques, researchers can address critical challenges in modeling, ensuring that solutions are not only data-driven but also consistent with real-world physical phenomena. This integrative approach is poised to drive breakthroughs across fields like fluid dynamics, material science, quantum mechanics, and beyond, enabling the creation of more accurate and reliable digital twins and simulation frameworks.