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1. Learning partial differential equations for biological transport models from noisy spatio-temporal data

   https://doi.org/10.1098/rspa.2019.0800  

   Lagergren, John H.; Nardini, John T.; Michael Lavigne, G.; Rutter, Erica M.; Flores, Kevin B. (February 2020, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   We investigate methods for learning partial differential equation (PDE) models from spatio-temporal data under biologically realistic levels and forms of noise. Recent progress in learning PDEs from data have used sparse regression to select candidate terms from a denoised set of data, including approximated partial derivatives. We analyse the performance in using previous methods to denoise data for the task of discovering the governing system of PDEs. We also develop a novel methodology that uses artificial neural networks (ANNs) to denoise data and approximate partial derivatives. We test the methodology on three PDE models for biological transport, i.e. the advection–diffusion, classical Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) and nonlinear Fisher–KPP equations. We show that the ANN methodology outperforms previous denoising methods, including finite differences and both local and global polynomial regression splines, in the ability to accurately approximate partial derivatives and learn the correct PDE model. 

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2. Front location determines convergence rate to traveling waves

   https://doi.org/10.4171/AIHPC/120  

   An, Jing; Henderson, Christopher; Ryzhik, Lenya (May 2024, Annales de l'Institut Henri Poincaré C, Analyse non linéaire)

   We propose a novel method for establishing the convergence rates of solutions to reaction–diffusion equations to traveling waves. The analysis is based on the study of the traveling wave shape defect function introduced in An et. al. [Arch. Ration. Mech. Anal. 247 (2023), no. 5, article no. 88]. It turns out that the convergence rate is controlled by the distance between thephantom front locationfor the shape defect function and the true front location of the solution. Curiously, the convergence to a traveling wave has a pulled nature, regardless of whether the traveling wave itself is of pushed, pulled, or pushmi-pullyu type. In addition to providing new results, this approach dramatically simplifies the proof in the Fisher–KPP case and gives a unified, succinct explanation for the known algebraic rates of convergence in the Fisher–KPP case and the exponential rates in the pushed case. 

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3. Langmuir Mixing Schemes Based on a Modified K‐Profile Parameterization

   https://doi.org/10.1029/2024MS004729  

   Wang, Peng; McWilliams, James C; Yuan, Jianguo; Liang, Jun‐Hong (April 2025, Journal of Advances in Modeling Earth Systems)

   Abstract Langmuir turbulence, a dominant process in the ocean surface boundary layer, drives substantial vertical mixing that influences temperature, salinity, mixed layer depth, and biogeochemical tracer distributions. While direct resolution of Langmuir turbulence in ocean and climate models remains computationally prohibitive, its effects are commonly parameterized, frequently within established turbulent mixing frameworks like the K‐profile parameterization (KPP). This study utilizes a modified KPP that determines boundary layer depth through an integral criterion, diverging from the conventional KPP's dependence on the bulk Richardson number. The modified KPP demonstrates markedly lower sensitivity to model vertical resolution than its conventional counterpart. Building upon this modified KPP framework, we introduce an innovative parameterization scheme for Langmuir mixing effects. We evaluate the performance of this new scheme against existing approaches using a one‐dimensional (1D) column model across four different scenarios, incorporating validation against both large eddy simulation (LES) results and field measurements. Our analysis reveals that the new Langmuir mixing scheme, explicitly designed for the modified KPP framework, performs competitively while maintaining reduced sensitivity to vertical resolution. 

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4. Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity

   https://doi.org/10.3934/dcdss.2021146  

   Faye, Grégory; Giletti, Thomas; Holzer, Matt (January 2022, Discrete and Continuous Dynamical Systems - S)

   We determine the asymptotic spreading speed of the solutions of a Fisher-KPP reaction-diffusion equation, starting from compactly supported initial data, when the diffusion coefficient is a fixed bounded monotone profile that is shifted at a given forcing speed and satisfies a general uniform ellipticity condition. Depending on the monotonicity of the profile, we are able to characterize this spreading speed as a function of the forcing speed and the two linear spreading speeds associated to the asymptotic problems at \begin{document}$$ x = \pm \infty $$\end{document}. Most notably, when the profile of the diffusion coefficient is increasing we show that there is an intermediate range for the forcing speed where spreading actually occurs at a speed which is larger than the linear speed associated with the homogeneous state around the position of the front. We complement our study with the construction of strictly monotone traveling front solutions with strong exponential decay near the unstable state when the profile of the diffusion coefficient is decreasing and in the regime where the forcing speed is precisely the selected spreading speed. 

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5. The K‐Profile Parameterization Augmented by Deep Neural Networks (KPP_DNN) in the General Ocean Turbulence Model (GOTM)

   https://doi.org/10.1029/2024MS004405  

   Yuan, Jianguo; Liang, Jun‐Hong; Chassignet, Eric P; Zavala‐Romero, Olmo; Wan, Xiaoliang; Cronin, Meghan F (September 2024, Journal of Advances in Modeling Earth Systems)

   Abstract This study utilizes Deep Neural Networks (DNN) to improve the K‐Profile Parameterization (KPP) for the vertical mixing effects in the ocean's surface boundary layer turbulence. The deep neural networks were trained using 11‐year turbulence‐resolving solutions, obtained by running a large eddy simulation model for Ocean Station Papa, to predict the turbulence velocity scale coefficient and unresolved shear coefficient in the KPP. The DNN‐augmented KPP schemes (KPP_DNN) have been implemented in the General Ocean Turbulence Model (GOTM). The KPP_DNN is stable for long‐term integration and more efficient than existing variants of KPP schemes with wave effects. Three different KPP_DNN schemes, each differing in their input and output variables, have been developed and trained. The performance of models utilizing the KPP_DNN schemes is compared to those employing traditional deterministic first‐order and second‐moment closure turbulent mixing parameterizations. Solution comparisons indicate that the simulated mixed layer becomes cooler and deeper when wave effects are included in parameterizations, aligning closer with observations. In the KPP framework, the velocity scale of unresolved shear, which is used to calculate ocean surface boundary layer depth, has a greater impact on the simulated mixed layer than the magnitude of diffusivity does. In the KPP_DNN, unresolved shear depends not only on wave forcing, but also on the mixed layer depth and buoyancy forcing. 

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