Locally-symplectic neural networks for learning volume-preserving dynamics

• URL: http://arxiv.org/abs/2109.09151v1
• Date: Sun, 19 Sep 2021 15:58:09 GMT
• Title: Locally-symplectic neural networks for learning volume-preserving dynamics
• Authors: J\=anis Baj\=ars
• Abstract summary: We propose locally-symplectic neural networks LocSympNets for learning volume-preserving dynamics. The construction of LocSympNets stems from the theorem of local Hamiltonian description of the vector field of a volume-preserving dynamical system.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose locally-symplectic neural networks LocSympNets for learning volume-preserving dynamics. The construction of LocSympNets stems from the theorem of local Hamiltonian description of the vector field of a volume-preserving dynamical system and the splitting methods based on symplectic integrators. Modified gradient modules of recently proposed symplecticity-preserving neural networks SympNets are used to construct locally-symplectic modules, which composition results in volume-preserving neural networks. LocSympNets are studied numerically considering linear and nonlinear dynamics, i.e., semi-discretized advection equation and Euler equations of the motion of a free rigid body, respectively. LocSympNets are able to learn linear and nonlinear dynamics to high degree of accuracy. When learning a single trajectory of the rigid body dynamics LocSympNets are able to learn both invariants of the system with absolute relative errors below 1% in long-time predictions and produce qualitatively good short-time predictions, when the learning of the whole system from randomly sampled data is considered.

Related papers

• Enhancing lattice kinetic schemes for fluid dynamics with Lattice-Equivariant Neural Networks [79.16635054977068]
  We present a new class of equivariant neural networks, dubbed Lattice-Equivariant Neural Networks (LENNs) Our approach develops within a recently introduced framework aimed at learning neural network-based surrogate models Lattice Boltzmann collision operators. Our work opens towards practical utilization of machine learning-augmented Lattice Boltzmann CFD in real-world simulations.
  arXiv  Detail & Related papers  (2024-05-22T17:23:15Z)
• Applications of Machine Learning to Modelling and Analysing Dynamical Systems [0.0]
  We propose an architecture which combines existing Hamiltonian Neural Network structures into Adaptable Symplectic Recurrent Neural Networks. This architecture is found to significantly outperform previously proposed neural networks when predicting Hamiltonian dynamics. We show that this method works efficiently for single parameter potentials and provides accurate predictions even over long periods of time.
  arXiv  Detail & Related papers  (2023-07-22T19:04:17Z)
• How neural networks learn to classify chaotic time series [77.34726150561087]
  We study the inner workings of neural networks trained to classify regular-versus-chaotic time series. We find that the relation between input periodicity and activation periodicity is key for the performance of LKCNN models.
  arXiv  Detail & Related papers  (2023-06-04T08:53:27Z)
• Constraining Chaos: Enforcing dynamical invariants in the training of recurrent neural networks [0.0]
  We introduce a novel training method for machine learning based forecasting methods for chaotic dynamical systems. The training enforces dynamical invariants--such as the Lyapunov exponent spectrum and fractal dimension--in the systems of interest, enabling longer and more stable forecasts when operating with limited data.
  arXiv  Detail & Related papers  (2023-04-24T00:33:47Z)
• ConCerNet: A Contrastive Learning Based Framework for Automated Conservation Law Discovery and Trustworthy Dynamical System Prediction [82.81767856234956]
  This paper proposes a new learning framework named ConCerNet to improve the trustworthiness of the DNN based dynamics modeling. We show that our method consistently outperforms the baseline neural networks in both coordinate error and conservation metrics.
  arXiv  Detail & Related papers  (2023-02-11T21:07:30Z)
• Decomposed Linear Dynamical Systems (dLDS) for learning the latent components of neural dynamics [6.829711787905569]
  We propose a new decomposed dynamical system model that represents complex non-stationary and nonlinear dynamics of time series data. Our model is trained through a dictionary learning procedure, where we leverage recent results in tracking sparse vectors over time. In both continuous-time and discrete-time instructional examples we demonstrate that our model can well approximate the original system.
  arXiv  Detail & Related papers  (2022-06-07T02:25:38Z)
• Learn Like The Pro: Norms from Theory to Size Neural Computation [3.848947060636351]
  We investigate how dynamical systems with nonlinearities can inform the design of neural systems that seek to emulate them. We propose a Learnability metric and quantify its associated features to the near-equilibrium behavior of learning dynamics. It reveals exact sizing for a class of neural networks with multiplicative nodes that mimic continuous- or discrete-time dynamics.
  arXiv  Detail & Related papers  (2021-06-21T20:58:27Z)
• Constrained Block Nonlinear Neural Dynamical Models [1.3163098563588727]
  Neural network modules conditioned by known priors can be effectively trained and combined to represent systems with nonlinear dynamics. The proposed method consists of neural network blocks that represent input, state, and output dynamics with constraints placed on the network weights and system variables. We evaluate the performance of the proposed architecture and training methods on system identification tasks for three nonlinear systems.
  arXiv  Detail & Related papers  (2021-01-06T04:27:54Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)
• Liquid Time-constant Networks [117.57116214802504]
  We introduce a new class of time-continuous recurrent neural network models. Instead of declaring a learning system's dynamics by implicit nonlinearities, we construct networks of linear first-order dynamical systems. These neural networks exhibit stable and bounded behavior, yield superior expressivity within the family of neural ordinary differential equations.
  arXiv  Detail & Related papers  (2020-06-08T09:53:35Z)
• Training End-to-End Analog Neural Networks with Equilibrium Propagation [64.0476282000118]
  We introduce a principled method to train end-to-end analog neural networks by gradient descent. We show mathematically that a class of analog neural networks (called nonlinear resistive networks) are energy-based models. Our work can guide the development of a new generation of ultra-fast, compact and low-power neural networks supporting on-chip learning.
  arXiv  Detail & Related papers  (2020-06-02T23:38:35Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.