Anaya Nolan
Neural networks are increasingly being used in the physical sciences to discover physical laws from data. While neural networks have been successful at solving concrete physics problems, their use as a general-purpose tool for scientific discovery is still in its infancy. Machine learning systems often rely on large and complex neural networks that can struggle to find the simplest explanation for an event or observation, a concept called parsimony. However, enforcing parsimony through stochastic optimization enables these neural networks to balance simplicity with accuracy, effectively discovering physical laws from data.
| Characteristics | Values |
|---|---|
| Use Case | Discovering physical laws |
| Data Source | Papers on Newton's second law of motion and the Lindemann melting law |
| Techniques | Stochastic optimization, symbolic regression, process-based models, GNNs |
| Benefits | Improved accuracy, reduced data requirements, Faster discovery of physical laws |
| Challenges | Mismatch between theory and measurements, measurement noise, complex secondary physical mechanisms |
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What You'll Learn
Neural networks and symbolic regression
Neural networks, on the other hand, have achieved remarkable accuracy in image recognition and natural language processing tasks. They have been applied to learn the entire analytic model, including its structure and coefficients, using gradient-based optimization algorithms. While neural networks are often seen as black-box models that are challenging to interpret and may not extrapolate well, they can be combined with symbolic regression techniques to overcome these limitations.
One such approach is the Equation Learner (EQL) network, which integrates neural network-based symbolic regression with other deep learning architectures. The EQL-based architecture has demonstrated superior extrapolation capabilities compared to standard neural network models, making it a promising tool for scientific exploration and discovery. This network can be trained end-to-end through backpropagation, allowing it to learn the form of various functions and perform symbolic regression effectively.
Additionally, parsimonious neural networks (PNNs) combine neural networks with evolutionary optimization to balance accuracy with parsimony, a concept that focuses on finding the simplest explanation for an event or observation. By enforcing parsimony, researchers have successfully trained neural networks to discover physical laws, such as Newton's second law of motion and the Lindemann melting law. This approach has also been applied to predict the melting temperature of materials and understand particle dynamics, showcasing the versatility and interpretability of PNNs.
In conclusion, neural networks and symbolic regression techniques, when used together, offer a powerful framework for scientific discovery and understanding complex data. By leveraging the strengths of both methods, researchers can develop models that are accurate, interpretable, and capable of generalizing beyond the training data. The combination of neural networks and symbolic regression holds great potential for accelerating the discovery of physical laws and enhancing our understanding of the natural world.
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Parsimonious neural networks
Neural networks are often too large and complex to easily discover the simplest explanation of an event or observation, a concept called parsimony. Parsimonious neural networks (PNNs) are a technique that enforces parsimony on neural networks through "stochastic optimization". This enables them to better balance simplicity with accuracy, effectively teasing out interesting physics from data.
PNNs combine neural networks with evolutionary optimization to find models that balance accuracy with parsimony. The power and versatility of the approach are demonstrated by developing models for classical mechanics and to predict the melting temperature of materials from fundamental properties. In the first example, the resulting PNNs are easily interpretable as Newton’s second law, expressed as a non-trivial time integrator that exhibits time-reversibility and conserves energy, where parsimony is critical to extract underlying symmetries from the data. In the second case, the PNNs not only find the celebrated Lindemann melting law, but also new relationships that outperform it.
PNNs have also been used to learn, solely from observational data, the dynamics of a particle under a highly non-linear potential. This is done by using prior knowledge of the underlying physics to guide the model, often as numerical constraints, or by using the underlying physics to numerically solve equations with variables predicted by the ML algorithms. Symbolic regression methods, enhanced by matching partial derivatives to identify invariants, have also been used to learn physics directly from data, without prior knowledge.
The idea behind PNNs is that imposing parsimony (e.g., limiting adjustable parameters and favoring linear correlations between variables) will require the resulting model to be easily interpretable and pull out the problem’s symmetries. Compared to a flexible feed-forward neural network, the resulting PNN lends itself to interpretation (like Newton’s laws) and provides a substantially more accurate description of the particle dynamics when applied iteratively. The resulting PNNs are energy efficient and time-reversible, which means they learn non-trivial symmetries that are implicit in the data but not explicitly presented.
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Process-based models
In the context of physics-informed neural networks (PINNs), process-based models play a crucial role in embedding the knowledge of physical laws within a given dataset. PINNs act as universal function approximators and can be described by partial differential equations (PDEs). They enhance the information content of the available data, enabling the learning algorithm to generalize well even with limited training examples. PINNs have proven effective in solving inverse problems within differential equations and have found applications across science, engineering, economics, and more.
While process-based models offer advantages, they also have certain limitations. For instance, they may struggle with complex problems that require handling high-dimensional data. This is where symbolic regression comes into play, as it excels at generalization and can complement deep learning models. By combining symbolic regression with neural networks, researchers aim to improve the interpretability and generalization capabilities of the models, making them more effective in discovering physical laws from data.
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Machine learning and physics-based modelling
Machine learning (ML) and artificial intelligence (AI) algorithms are now being used to automate the discovery of physics principles and governing equations from measurement data. However, there are challenges to this approach, such as the need for a discrepancy model to account for the mismatch between theory and measurements. This is where physics-based modelling comes in.
Physics-based modelling is a process-based approach that uses domain knowledge to specify a library of relationships or functions between variables. These models can be used to support irrigation water management, providing the timing and amount of water needed, increasing productivity and reducing water use. For example, a process-based model was proposed to predict the water needs for different cranberry farm operations.
Machine learning, on the other hand, has the advantage of being easy to implement and requiring less computational effort for training. It has been used to develop models for classical mechanics and to predict the melting temperature of materials. However, one challenge with machine learning systems is that they often become too large and complex to easily discover the simplest explanation of an event or observation, a concept called parsimony.
To address this, researchers have proposed parsimonious neural networks (PNNs) that combine neural networks with evolutionary optimization to find models that balance accuracy with parsimony. These PNNs have been used to learn particle dynamics and predict the melting temperature of materials. By enforcing parsimony through stochastic optimization, neural networks can better balance simplicity with accuracy, effectively teasing out interesting physics from data.
In conclusion, both machine learning and physics-based modelling have their strengths and weaknesses. Machine learning is a powerful tool that can improve scientific research, but issues such as integrating it with physics-based modelling need to be addressed. Physics-based modelling provides accurate predictions but may require more computational effort. By combining these approaches, scientists can discover physical laws faster and improve the accuracy of their models.
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Neural networks and generalisation
Neural networks are being used in physics to discover physical laws from data. However, one of the challenges in this field is the generalisation of the neural network. Generalisation refers to a model's ability to react to new data, i.e. its ability to digest new data and make accurate predictions. This is important because a model that has been trained too well on training data may struggle to generalise.
Generalisation is ensured by testing the neural network on data it has not been trained on. If the network performs well on this new data, it has generalised well. However, this is one of the most frustrating tasks in deep learning due to the phenomenon of overfitting. Overfitting occurs when a neural network memorises certain data and fails to form a general understanding. For example, if a neural network has been trained on 10 breeds of dogs, it may refuse to classify the other 2 breeds as dogs.
There are ways to combat overfitting. One popular method is to use dropout, which encourages the network to utilise every node and avoid dependencies on small clusters of nodes. This makes training harder, but it is easier to cut training when the network has generalised well. Another solution is to simply add more data. There are also certain aspects of neural networks that can be controlled to prevent overfitting, such as the number of parameters.
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Frequently asked questions
One challenge is the need for a large amount of data, which can be expensive and time-consuming to acquire, especially in fields like materials science. Another challenge is the inevitable mismatch between theory and measurements, which can lead to erroneous models if not addressed through a discrepancy model.
Neural networks can be combined with evolutionary optimization techniques like stochastic optimization to balance accuracy and parsimony, allowing the discovery of underlying physical laws from observational data. This approach has been successfully applied to classical mechanics and the prediction of material properties.
Neural networks offer a general-purpose tool for scientific discovery, providing insights and improving generalizability. They can extract relevant features from data, identify underlying symmetries, and make predictions based on conservation laws, all without prior assumptions about the system.
Researchers from Purdue University demonstrated the effectiveness of neural networks in discovering physical laws. They trained neural networks using data from papers on Newton's second law of motion and the Lindemann melting law. After simulations, the networks produced models that successfully demonstrated these physical laws.
