The law of conservation of dynamic systems is no mystery to artificial intelligence

(Nanowerk News) Many real-world systems, from climate systems to the physical mechanics of robots, are governed by invariant quantities that arise from their underlying geometric structures. Modeling these systems using computer simulations is a key tool for understanding them (for weather forecasting, for example, or developing robotic propulsion). Collecting data for these systems is often possible, but making sense of that data to create models is a more challenging task.

Artificial intelligence has long been used to analyze system data with known conservation laws. However, in the real world, these laws are often unknown. To solve this problem, researchers from Osaka University developed an algorithm called FINDE, which finds not only a number of known laws, but also unknown laws, which can lead to scientific discoveries. Figure 1 From left to right, the trajectory of the double pendulum, the predicted results using conventional artificial neural networks, and those using the method proposed in this study. Conventional artificial neural networks cannot find that the length of the pendulum rod remains constant, and errors accumulate over time. The proposed method finds and preserves this law. (Image: Takashi Matsubara, Takaharu Yaguchi)

FINDE assumes quantity invariant and projects dynamics onto a manifold tangent space defined by quantity. This allows FINDE to study the dynamics and magnitudes of these invariants together, respecting the laws of conservation. Real-world systems are continuous systems, but numerical computer algorithms can only model discrete systems, which can introduce errors and ultimately reduce simulation accuracy. Therefore, FINDE also uses discrete gradients, which are defined in discrete time, to determine discrete pairs of tangents, and can then be used without accumulating errors.

“Because FINDE studies and preserves the conservation laws of a system,” said Prof. Takashi Matsubara, first author of the study, “our computer simulations will be more accurate in the long run.”

However, FINDE does more than reduce modeling errors. If the laws governing a system are unknown, the number of laws may also be unknown. FINDE can be used to determine this number by testing the accuracy of the model with different legal numbers. “This reveals additional information about the underlying structure of a system,” said Prof. Matsubara, and it can lead to scientific discoveries about the system under study. Figure 2 From left to right, the trajectory of the two-object system, the predicted results using a conventional artificial neural network, and those using the method proposed in this study. In contrast to conventional neural networks, the proposed method is able to find and maintain laws that save momentum. (Image: Takashi Matsubara, Takaharu Yaguchi)

The researchers tested FINDE’s ability to discover and model systems on various data sets. Two-body gravitational systems, shallow water waves, double pendulums, and electrical circuit models of biological neurons are all considered. FINDE can find and maintain the first integral that arises from the conservation law that governs this system in all cases. As a result, FINDE is expected to contribute to the acceleration and sophistication of computer-aided physics and engineering simulation in the future.

The article, “FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantities,” will be presented at the 2023 International Conference on Representation of Learning.