More Like this

1. Online Weak-form Sparse Identification of Partial Differential Equations

   Messenger, Daniel; Dall’Anese, Emiliano; Bortz, David (July 2022, Proceedings of Mathematical and Scientific Machine Learning, PMLR)

   This paper presents an online algorithm for identification of partial differential equations (PDEs) based on the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy). The algorithm is online in a sense that if performs the identification task by processing solution snapshots that arrive sequentially. The core of the method combines a weak-form discretization of candidate PDEs with an online proximal gradient descent approach to the sparse regression problem. In particular, we do not regularize the ℓ0 -pseudo-norm, instead finding that directly applying its proximal operator (which corresponds to a hard thresholding) leads to efficient online system identification from noisy data. We demonstrate the success of the method on the Kuramoto-Sivashinsky equation, the nonlinear wave equation with time-varying wavespeed, and the linear wave equation, in one, two, and three spatial dimensions, respectively. In particular, our examples show that the method is capable of identifying and tracking systems with coefficients that vary abruptly in time, and offers a streaming alternative to problems in higher dimensions. 

   more » « less
2. Asymptotic consistency of the WSINDy algorithm in the limit of continuum data

   https://doi.org/10.1093/imanum/drae086  

   Messenger, Daniel_A; Bortz, David_M (December 2024, IMA Journal of Numerical Analysis)

   Abstract In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models that includes the Navier–Stokes, Kuramoto–Sivashinsky and Sine–Gordon equations. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that, in general, the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level exceeds a critical threshold $$\sigma _{c}$$. We provide explicit bounds on $$\sigma _{c}$$ in the case of Gaussian white noise and we explicitly characterize the spurious terms that arise in the case of trigonometric and/or polynomial libraries. Furthermore, we show that, if the data is suitably denoised (a simple moving average filter is sufficient), then asymptotic consistency is recovered for models with locally-Lipschitz, polynomial-growth nonlinearities. Our results reveal important aspects of weak-form equation learning, which may be used to improve future algorithms. We demonstrate our findings numerically using the Lorenz system, the cubic oscillator, a viscous Burgers-growth model and a Kuramoto–Sivashinsky-type high-order PDE. 

   more » « less
3. Robust Sparse Online Learning for Data Streams with Streaming Features

   https://doi.org/10.1137/1.9781611978032.21  

   Chen, Zhong; He, Yi; Wu, Di; Zhan, Huixin; Sheng, Victor; Zhang, Kun (January 2024, Proceedings of the 2024 SIAM International Conference on Data Mining (SDM))

   Sparse online learning has received extensive attention during the past few years. Most of existing algorithms that utilize ℓ1-norm regularization or ℓ1-ball projection assume that the feature space is fixed or changes by following explicit constraints. However, this assumption does not always hold in many real applications. Motivated by this observation, we propose a new online learning algorithm tailored for data streams described by open feature spaces, where new features can be occurred, and old features may be vanished over various time spans. Our algorithm named RSOL provides a strategy to adapt quickly to such feature dynamics by encouraging sparse model representation with an ℓ1- and ℓ2-mixed regularizer. We leverage the proximal operator of the ℓ1,2-mixed norm and show that our RSOL algorithm enjoys a closed-form solution at each iteration. A sub-linear regret bound of our proposed algorithm is guaranteed with a solid theoretical analysis. Empirical results benchmarked on nine streaming datasets validate the effectiveness of the proposed RSOL method over three state-of-the-art algorithms. Keywords: online learning, sparse learning, streaming feature selection, open feature spaces, ℓ1,2 mixed norm 

   more » « less
4. On the Wave Turbulence Theory for the Nonlinear Schrödinger Equation with Random Potentials

   https://doi.org/https://doi.org/10.3390/e21090823  

   Nazarenko, Sergy; Soffer, Avy; Tran, Minh-Binh (August 2019, Entropy)

   We derive new kinetic and a porous medium equations from the nonlinear Schrödinger equation with random potentials. The kinetic equation has a very similar form compared to the four-wave turbulence kinetic equation in the wave turbulence theory. Moreover, we construct a class of self-similar solutions for the porous medium equation. These solutions spread with time, and this fact answers the “weak turbulence” question for the nonlinear Schrödinger equation with random potentials. We also derive Ohm’s law for the porous medium equation. 

   more » « less
5. Alternative formulation of weak magnetohydrodynamic turbulence theory

   https://doi.org/10.1063/5.0097084  

   Yoon, Peter H.; Ziebell, Luiz F.; Choe, Gwangson (November 2022, Physics of Plasmas)

   In a recent paper [P. H. Yoon and G. Choe, Phys. Plasmas 28, 082306 (2021)], the weak turbulence theory for incompressible magnetohydrodynamics is formulated by employing the method customarily applied in the context of kinetic weak plasma turbulence theory. Such an approach simplified certain mathematical procedures including achieving the closure relationship. The formulation in the above-cited paper starts from the equations of incompressible magnetohydrodynamic (MHD) theory expressed via Elsasser variables. The derivation of nonlinear wave kinetic equation therein is obtained via a truncated solution at the second-order of iteration following the standard practice. In the present paper, the weak MHD turbulence theory is alternatively formulated by employing the pristine form of incompressible MHD equation rather than that expressed in terms of Elsasser fields. The perturbative expansion of the nonlinear momentum equation is carried out up to the third-order iteration rather than imposing the truncation at the second order. It is found that while the resulting wave kinetic equation is identical to that obtained in the previous paper cited above, the third-order nonlinear correction plays an essential role for properly calculating derived quantities such as the total and residual energies. 

   more » « less