Kara Sears
Machine learning (ML) and artificial intelligence (AI) are increasingly being used to automate the discovery of physics principles and governing equations from measurement data. Neural networks are being used to discover physical laws from data, with researchers from Purdue University finding a way to use machine learning to reduce the time it takes to identify physical laws from decades to just a few days. Parsimonious neural networks (PNNs) are also being used to find models that balance accuracy with parsimony, and have been used to develop models for classical mechanics and to predict the melting temperature of materials. Symbolic regression is another method that is being used to discover the laws of physics, where a deep neural network is transformed into a simplified mathematical equation.
| Characteristics | Values |
|---|---|
| Use of neural networks | Discovering physical concepts |
| Neural network architecture | Modelled after the human physical reasoning process |
| Neural network function | Anticipate targets and split them into tiny internal functions |
| Neural network benefits | Can be used to solve concrete physics problems |
| Neural network limitations | Often too large and complex to easily explain an event or observation |
| Neural network improvements | Combining with evolutionary optimization to improve accuracy |
| Neural network applications | Producing models that demonstrate physical laws, such as Newton's second law of motion |
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What You'll Learn
Using machine learning to discover physical laws from data
Machine learning (ML) and artificial intelligence (AI) algorithms are being used to automate the discovery of physics principles and governing equations from measurement data. This has been applied to revisit classic problems in physics, such as the motion of falling objects and their relationship with gravitation. However, one of the challenges of using ML to discover universal physical laws is the inevitable mismatch between theory and measurements, which can lead to erroneous models.
ML models often struggle with learning new physics and explaining predictions, and their performance is usually better when applied to data similar to what they have already seen. They tend to excel at interpolation but are often poor extrapolators. To address this, researchers at Purdue University have developed a technique called "stochastic optimization" to enforce parsimony on neural networks, enabling them to balance simplicity with accuracy and extract meaningful physics from data.
The Purdue researchers trained parsimonious neural networks using data from Newton's second law of motion and the Lindemann melting law. Their approach not only interpreted these physical laws but also optimized the Lindemann melting law to be simpler and more accurate. This demonstrates the potential for ML to accelerate the discovery of physical laws, reducing the time from decades to just a few days.
Graph neural networks are another specialized class of neural networks that have proven effective at learning basic physics simulators from measurement data and directly from videos. Other deep learning methods have also been proposed, such as building neural networks that respect given physical laws, discovering parameters in non-linear partial differential equations with limited data, and simultaneously approximating the solution and non-linear dynamics of these equations. These advancements in ML and data science are providing new opportunities for scientific exploration and discovery.
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Neural networks and stochastic optimization
Neural networks are a well-utilized branch of machine learning that has already replaced humans in solving many problems. Neural networks can be trained using optimization methods, which improve their accuracy without increasing training time.
Stochastic optimization is a learning algorithm for neural networks that performs random optimization in the weight space. It does not make any assumptions about transfer functions of individual neurons and does not depend on a functional form of a performance measure. The algorithm uses a random step of varying size to adapt weights. The average size of the step decreases during learning. Large steps enable the algorithm to jump over local maxima/minima, while small steps ensure convergence in a local area.
Solis and Wets proposed a modified random optimization method to find the global minimum of the error function in a small number of steps. This method converges faster than Matyas' method. The algorithm performs a random search in weight space with steps of varying sizes. Small steps enable the algorithm to perform a fine weight tuning in a local area, while large steps guarantee a search in the whole weight space.
Other optimization algorithms include fractional-order, bilevel, and gradient-free optimizers, which can replace classical gradient-based optimizers. Fast gradient methods have some practical gains in the stochastic case, where they partially mimic their exact gradient counterparts.
Purdue University researchers have used stochastic optimization to train parsimonious neural networks to discover physical laws from data. This approach combines neural networks with evolutionary optimization to find models that balance accuracy with parsimony. The researchers trained the neural networks using data from papers on Newton’s second law of motion and the Lindemann melting law.
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Graph neural networks
In computer science, a graph is a data structure consisting of two components: nodes (vertices) and edges. GNNs can be used to determine the labelling of samples (represented as nodes) by looking at the labels of their neighbours. This is known as node classification. Another application of GNNs is graph classification, where the task is to classify the whole graph into different categories.
GNNs have been shown to be successful in learning physics problems. The message function of a GNN is analogous to a force, while the node update function is analogous to Newton's law of motion. Introducing the law of momentum conservation in GNN models can improve the efficiency and stability of learning, allowing convergence to better models with fewer training steps.
A machine learning method called the "Graph-based Physics Engine" (GPE) uses GNNs to efficiently model the physical dynamics of different substances in a wide variety of scenarios. GPE can generalize to materials with different properties not seen in the training set and perform well from single-step predictions to multi-step roll-out simulations.
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Symbolic regression
One notable algorithm in symbolic regression is the AI Feynman algorithm, developed by Silviu-Marian Udrescu and Max Tegmark. This algorithm trains a neural network to represent the mystery function and then runs tests to break down the problem into smaller, more manageable parts. AI Feynman also transforms the inputs and outputs of the function to produce a new function that can be solved using other techniques.
In the context of discovering physical laws, machine learning systems, including neural networks, have been used to automate the process. Techniques such as parsimonious neural networks (PNNs) combine neural networks with evolutionary optimization to balance accuracy and simplicity. This approach has been successful in learning particle dynamics and predicting the melting temperature of materials.
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Deep neural networks
Machine learning systems can use deep neural networks to discover physical laws from data. Deep neural networks are a type of machine learning model that mimics the human brain's ability to process information and recognise complex patterns. They are composed of multiple layers of nodes (neurons) that receive input from other layers and produce an output until a final result is reached. The "deep" in deep neural networks refers to the multiple hidden layers between the input and output layers. These hidden layers allow the network to learn from previous experiences and make predictions based on patterns recognised in large datasets.
In addition to discovering physical laws, deep neural networks have been applied in various fields, including medical image analysis, computer vision, speech recognition, natural language processing, and drug design. They have also been used to develop AI agents, image recognition systems, and voice assistants.
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Frequently asked questions
Discovering physical laws from data is challenging because of the inevitable mismatch between theory and measurements. For example, measurement noise and complex secondary physical mechanisms, like unsteady fluid drag forces, can obscure the underlying law of gravitation, leading to an erroneous model.
Neural networks can be combined with evolutionary optimisation techniques such as stochastic optimisation to find models that balance accuracy with parsimony. This approach has been used to develop models for classical mechanics and to predict the melting temperature of materials from fundamental properties.
Neural networks can be used to discover physical laws from data without prior knowledge of the underlying physics. They can also be used to parametrize and solve differential equations such as Navier Stokes and Hamilton’s equations of motion.
