Machine Learning with Physics Knowledge for Prediction: A Survey (2408.09840v1)

Abstract: This survey examines the broad suite of methods and models for combining machine learning with physics knowledge for prediction and forecast, with a focus on partial differential equations. These methods have attracted significant interest due to their potential impact on advancing scientific research and industrial practices by improving predictive models with small- or large-scale datasets and expressive predictive models with useful inductive biases. The survey has two parts. The first considers incorporating physics knowledge on an architectural level through objective functions, structured predictive models, and data augmentation. The second considers data as physics knowledge, which motivates looking at multi-task, meta, and contextual learning as an alternative approach to incorporating physics knowledge in a data-driven fashion. Finally, we also provide an industrial perspective on the application of these methods and a survey of the open-source ecosystem for physics-informed machine learning.

• The paper systematically surveys methods for integrating physics knowledge into ML models to enhance prediction, particularly for partial differential equations.
• It analyzes two main approaches: embedding physics directly into neural architectures and using physics-informed loss functions to constrain learning.
• The survey also evaluates industrial applications and open-source tools, highlighting future directions for robust, hybrid prediction models.

Machine Learning with Physics Knowledge for Prediction: A Survey

Overview

The paper "Machine Learning with Physics Knowledge for Prediction: A Survey" systematically examines methodologies that integrate ML with physics knowledge for enhancing predictive capabilities, particularly for solving partial differential equations (PDEs). The survey delineates two primary methods: embedding physics intuition directly into model architectures and using physics knowledge to inform data-driven learning processes. The survey also explores pertinent industrial applications and evaluates open-source tools and datasets in this growing research area.

Embedding Physics Knowledge in Model Architectures

1. Physics-Inspired Neural Networks:
   • Neural ODEs and SDES aim to model continuous-time systems by representing differential equations as neural networks, which allows for predictions on irregular time intervals. Embedding physics knowledge directly into these architectures has enhanced model robustness and reduced data requirements.
   • Physics-Informed Neural Networks (PINNs) use automatic differentiation to embed PDE constraints directly into the learning process. While offering flexibility regarding domain complexity, inherent challenges include stabilization of the training process and ensuring the accuracy of the numerical approximations.
2. Physics-Constrained Loss Functions:
   • These approaches integrate physics-derived constraints into loss functions, reducing the search space for ML models by enforcing known physical laws. Methods discussed include PINNs, variational formulations, and data-driven penalization strategies that leverage large scientific datasets effectively.
3. Operator Learning:
   • Deep Operator Networks (DeepONets) and Neural Operators (NOs) capture mappings from input functions to output solutions, learning parameterized transformations inherent in physical processes. These methods promote discretization invariance and generalizability across different boundary conditions and PDE formulations.
4. Latent Variable Models:
   • Approaches like Koopman operator theory and non-linear latent variable models offer mechanisms to embed prior physical insights into dynamical systems, permitting effective extrapolation and robust forecasting, even when data is sparse.
5. Parameterizing Differential Equations:
   • Neural network-based parameterization of equations enables structured prediction by leveraging inductive biases, such as conservation laws and symmetries found in Hamiltonian and Lagrangian dynamics. This can help to maintain essential physical properties in the predictive models.

Data-Driven Learning Paradigms

1. Multitask Learning (MTL):
   • MTL techniques train models on multiple related tasks simultaneously, leveraging commonalities to improve model robustness and generalization. For example, multi-head PINNs have demonstrated effectiveness in learning across varied PDEs, benefiting from shared representations and task-specific adaptations.
2. Meta-Learning:
   • Meta-learning, exemplified by MAML and its derivatives, optimizes learning algorithms themselves. These techniques pre-train models to expedite fine-tuning on novel tasks, significantly enhancing prediction accuracy in complex, variable settings.
3. Neural Processes:
   • Neural processes combine meta-learning and Bayesian principles to rapidly adapt models with minimal new data, providing a flexible solution in dynamically varying scenarios. This approach is especially valuable in scenarios where quick updates and robustness to new conditions are critical.

Industrial Implications

The application of physics-informed machine learning spans numerous industries, where model reliability and interpretability are paramount. Examples include weather prediction, robotic control systems, and mechanical system diagnostics. A notable advantage is the ability to integrate data-driven prediction models into existing deterministic frameworks, offering enhanced accuracy and computational efficiency. Open-source projects like DeepXDE, Modulus, and the SciML suite provide valuable tools for researchers and practitioners, fostering collaboration and innovation.

Future Directions

Despite significant advancements, challenges remain, including stabilizing algorithms, enhancing scalability, and improving the convergence of hybrid physics-ML models. Future research may benefit from the continued development of foundation models for physical sciences, better uncertainty quantification methodologies, and integrating real-time adaptive learning. The industrial application highlights indicate a promising but cautious adoption, where hybridizing classical numerical approaches with modern ML techniques could yield the most practical results.

Closing Remarks

The integration of machine learning with domain-specific physics knowledge represents a substantial leap in predictive modeling. This survey highlights the varied approaches and underlying principles, offering a comprehensive guide for leveraging these methods effectively. Continued interdisciplinary collaboration and the development of open-source tools will be crucial in driving this field forward.