IDENTIFYING PHYSICAL LAW OF HAMILTONIAN SYS-TEMS VIA META-LEARNING

Abstract

Hamiltonian mechanics is an effective tool to represent many physical processes with concise yet well-generalized mathematical expressions. A well-modeled Hamiltonian makes it easy for researchers to analyze and forecast many related phenomena that are governed by the same physical law. However, in general, identifying a functional or shared expression of the Hamiltonian is very difficult. It requires carefully designed experiments and the researcher's insight that comes from years of experience. We propose that meta-learning algorithms can be potentially powerful data-driven tools for identifying the physical law governing Hamiltonian systems without any mathematical assumptions on the representation, but with observations from a set of systems governed by the same physical law. We show that a well meta-trained learner can identify the shared representation of the Hamiltonian by evaluating our method on several types of physical systems with various experimental settings.

1. INTRODUCTION

Hamiltonian mechanics, a reformulation of Newtonian mechanics, can be used to describe classical systems by focusing on modeling continuous-time evolution of system dynamics with a conservative quantity called Hamiltonian (Goldstein et al., 2002) . Interestingly, the formalism of the Hamiltonian provides both geometrically meaningful interpretation (Arnol'd et al., 2001) and efficient numerical schemes (Feng & Qin, 2010) representing the state of complex systems in phase space with symplectic structure. Although formalism was originally developed for classical mechanics, it has been applied to various fields of physics, such as fluid mechanics (Salmon, 1988) , statistical mechanics (Reichl, 1999) , and quantum mechanics (Sakurai & Commins, 1995) . While it has many useful mathematical properties, establishing an appropriate Hamiltonian of the unknown phenomena is a challenging problem. A Hamiltonian for a system can be modeled by a shared expression of the Hamiltonian and physical parameters. For instance, the Hamiltonian of an ideal pendulum is described as H = p 2 2ml 2 + mgl(1 -cos q) (shared expression), with mass m, pendulum length l, and gravity constant g (physical parameters), whereas q and p are the angle of the pendulum and the corresponding conjugate momentum (state of the system), respectively. Once an appropriate functional of the Hamiltonian is established from observing several pendulums, a new pendulum-like system can be readily recognized by adapting new physical parameters on the expression. Therefore, identifying an appropriate expression of the Hamiltonian is an important yet extremely difficult problem in most science and engineering areas where there still remain numerous unknown processes where it is even uncertain whether a closed-form solution or mathematically clear expression exists. In the recent era of deep learning, we can consider the use of learning-based algorithms to identify an appropriate expression of the Hamiltonian with sufficient data. To determine the Hamiltonian underlying the unknown physical process, the Hamiltonian should satisfy two fundamental conditions: (1) it should fit well on previously observed data or motions, (2) it should generalize well on newly observed data from new systems if the systems share the same physical law with previous ones. The first condition has been mitigated by explicitly incorporating symplectic structure or conservation laws on neural networks, called Hamiltonian neural networks (HNN) (Greydanus et al., 2019) for learning Hamiltonian dynamics. HNN and its variants have been shown to be effective in learning Figure 1 : There is a resemblance between meta-learning and identifying the physical laws of Hamiltonian. A hypothesized governing equation of Hamiltonian, usually corrected and established by evaluating many related systems, could be learned using meta-learning as a data-driven method (left). Then, a well-established Hamiltonian can be utilized to predict new system dynamics, which could be viewed as a meta-transfer process by a well-trained meta-learner (right). many useful properties of the Hamiltonian (Toth et al., 2020; Chen et al., 2020; Zhong et al., 2020a; Sanchez-Gonzalez et al., 2019; Jin et al., 2020) . In their experiments, it has been shown that HNN and its variants work well on learning conservation laws or continuous-time translational symmetry, enable the learning of complex systems stably by incorporating numerical integrators and generalize on multiple initial conditions or controls for the given system. However, there is limited work regarding a trained model that works well on totally new systems governed by the same physical law with novel physical parameters. To consider the second condition, we propose that meta-learning, which aims to train a model well generalized on novel data from observing a few examples, can be a potential key to learning a functional of Hamiltonian as a data-driven method. There have been several representative categories of meta-learning algorithms, such as the metric-based method (Snell et al., 2017; Sung et al., 2018 ), black-box method (Santoro et al., 2016; Bertinetto et al., 2019), and gradient-based method (Rusu et al., 2019; Flennerhag et al., 2020) . Among these methods, we especially focus on the gradientbased method, which is readily compatible with any differentiable model and flexibly applicable to a wide variety of learning problems (Finn et al., 2017; Xu et al., 2018; Hospedales et al., 2020) . One of the most successful algorithms of the gradient-based method is Model-Agnostic Meta-Learning (MAML) (Finn et al., 2017) , which consists of a task-specific adaptation process and a meta-optimization process. The key observations supporting its potential are the resemblance between these processes and the identification of the physical laws of the Hamiltonian. The schematic is shown in Figure 1 . The task-adaptation process, which adapts the initial model parameters to a task-specific train set, resembles the process of adapting hypothesized governing equations to observations of several physical systems. The meta-optimization process, which updates the initial model parameters by validating each task-specific adapted parameters to a task-specific test set, is similar to correcting the hypothesized governing equations by validating each system-specific Hamiltonian on new data from the corresponding physical systems. In addition, (Raghu et al., 2020) proposed that the recent success behind these meta-learning algorithms was due to providing qualitative shared representation across tasks rather than learning initial model parameters that encourage rapid adaptation (Finn et al., 2017) . This hypothesis may support our suggestion that a meta-learner can be efficient in identifying the shared representation of a Hamiltonian. From this point of view, we experiment on several types of physical systems to verify whether these meta-learning algorithms are beneficial to our desired learning problems. Our contributions are summarized as follows: • We formulate the problem of identifying the shared representation of unknown Hamiltonian as a meta-learning problem. • For learning to identify the Hamiltonian representations, we incorporate the HNN architecture on meta-learning algorithms. • After meta-training the meta-learner, we adapt the model on new systems by learning the data of partial observations and predict the dynamics of the systems as a vector field in phase space. • We evaluate our method on several types of physical systems to explore the efficiency of our methods with various experimental settings.

