WSINDy algorithm proves robust to noise in identifying differential equations.
problem Identifying differential equations from noisy data.
method Weak-form sparse identification of nonlinear dynamics (WSINDy) algorithm.
result WSINDy is asymptotically consistent for a wide class of models, including Navier-Stokes and Kuramoto-Sivashinsky equations.
This note introduces a regression technique for finding a class of nonlinear integro-differential operators from data. The method parametrizes the spatial operator with neural networks and Fourier transforms such that it can fit a class of nonlinear operators without needing a library of a priori selected operators. We…
ESNs with transfer learning predict long-term chaotic patterns in spatiotemporal dynamical systems.
problem Predicting long-term statistical patterns of spatiotemporally chaotic dynamical systems.
method Echo state networks (ESNs) with transfer learning.
result ESNs with transfer learning accurately predict long-term statistical properties of spatiotemporally chaotic PDEs.
MEDIDA discovers model errors in chaotic systems using sparse regression and data assimilation.
problem Model errors in chaotic systems lead to significant discrepancies between model predictions and real-world states.
method MEDIDA combines Bayesian sparse regression and data assimilation to estimate and interpret model errors from noisy observations.
result MEDIDA successfully identifies different types of model errors in the chaotic Kuramoto-Sivashinsky system.
The well-known Mori-Zwanzig theory tells us that model reduction leads to memory effect. For a long time, modeling the memory effect accurately and efficiently has been an important but nearly impossible task in developing a good reduced model. In this work, we explore a natural analogy between recurrent neural network…
FiniteNet uses a neural network to improve PDE solving methods.
problem Improving accuracy in solving time-dependent PDEs.
method Fully convolutional LSTM network trained on simulation data.
result Reduces error by a factor of 2 to 3 compared to baseline methods.
We develop a deep autoencoder architecture that can be used to find a coordinate transformation which turns a nonlinear PDE into a linear PDE. Our architecture is motivated by the linearizing transformations provided by the Cole-Hopf transform for Burgers equation and the inverse scattering transform for completely int…
Bayesian method learns PDEs from noisy data.
problem Discovering PDEs from noisy data.
method Combining variational Bayes and sparse linear regression.
result Proposes a new method to discover PDEs accurately.
Online algorithm identifies PDEs from noisy data snapshots.
problem Identifying PDEs from sequential solution snapshots.
method Combines weak-form discretization with online proximal gradient descent.
result Efficiently identifies and tracks systems with time-varying coefficients.
A lightweight framework improves convergence and stability of PINNs for complex PDEs.
problem Training instability and reduced accuracy in PINNs for complex PDEs.
method Adaptive curvature correction using secant information to optimize first-order optimizers.
result Consistent improvements in convergence speed, stability, and accuracy over standard optimizers.
While there is currently a lot of enthusiasm about "big data", useful data is usually "small" and expensive to acquire. In this paper, we present a new paradigm of learning partial differential equations from {\em small} data. In particular, we introduce \emph{hidden physics models}, which are essentially data-efficien…
In recent years, deep learning has proven to be a viable methodology for surrogate modeling and uncertainty quantification for a vast number of physical systems. However, in their traditional form, such models can require a large amount of training data. This is of particular importance for various engineering and scie…
A new method uses neural networks to improve POD-Galerkin models for complex systems.
problem Improving computational efficiency and accuracy in solving non-linear high-dimensional systems.
method Deep learning-based closure modeling using neural networks to approximate POD-Galerkin operators.
result The CD-ROM approach produces more accurate and stable models for complex systems.
This paper describes a method for learning low-dimensional approximations of nonlinear dynamical systems, based on neural-network approximations of the underlying Koopman operator. Extended Dynamic Mode Decomposition (EDMD) provides a useful data-driven approximation of the Koopman operator for analyzing dynamical syst…
We use standard deep neural networks to classify univariate time series generated by discrete and continuous dynamical systems based on their chaotic or non-chaotic behaviour. Our approach to circumvent the lack of precise models for some of the most challenging real-life applications is to train different neural netwo…
Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful re…
Generative models speed up complex system simulations.
problem Accurately forecasting the dynamics of complex systems at reduced cost.
method Generative Learning of Effective Dynamics (G-LED) using auto-regressive attention and Bayesian diffusion models.
result Generative models can accurately forecast complex system dynamics at lower computational cost.
A long-standing problem at the interface of artificial intelligence and applied mathematics is to devise an algorithm capable of achieving human level or even superhuman proficiency in transforming observed data into predictive mathematical models of the physical world. In the current era of abundance of data and advan…
Enhanced autoencoders improve ROMs for PDEs by capturing essential properties.
problem Autoencoders struggle to capture essential properties for accurate ROMs.
method Introduced symmetric Convolutional AutoEncoders (CAEs) that preserve manifold properties.
result Symmetric CAEs yield more accurate latent trajectories and robust models.
Model reduction methods aim to describe complex dynamic phenomena using only relevant dynamical variables, decreasing computational cost, and potentially highlighting key dynamical mechanisms. In the absence of special dynamical features such as scale separation or symmetries, the time evolution of these variables typi…
New PINN formulation respects causality for complex systems.
problem Existing PINNs fail to accurately simulate chaotic systems.
method Proposed a simple re-formulation of PINNs loss functions to respect physical causality.
result Significant accuracy improvements across chaotic systems.
TreeDOX predicts chaotic systems without hyperparameter tuning.
problem Forecasting chaotic systems requires hyperparameter tuning, limiting adoption.
method TreeDOX uses time delay overembedding and Extra-Trees Regressors.
result TreeDOX achieves state-of-the-art performance on chaotic systems.
Efficiently constructs sparse ROMs for high-dimensional data using causation entropy.
problem Creating effective reduced-order models for high-dimensional dynamical data.
method Uses causation entropy to identify important terms and construct ROMs with varying sparsity.
result Demonstrates the effectiveness of causation entropy in constructing sparse ROMs for chaotic systems with skewed statistics.
HFNO enhances interpretability of turbulent flows through parallel wavenumber bin processing.
problem Opaque inner workings of Fourier Neural Operators (FNOs) hinder physical interpretability.
method Introduces HFNO, a novel FNO-based architecture that processes wavenumber bins in parallel, enhancing interpretability.
result HFNO decomposes turbulent flows across various scales, enabling increased interpretability and multiscale modeling.
Develops theory for data-driven methods in dynamical systems.
problem Lack of analysis for data-driven methods in dynamical systems.
method Establishes existence of mapping and properties of operator learning architecture.
result Novel universal approximation theorems for smoothing and forecasting.
Novel autoencoder method approximates Koopman operator in low dimensions.
problem Challenges in approximating finite Koopman operators using data-driven methods.
method Mori-Zwanzig autoencoder (MZ-AE) for robust Koopman operator approximation.
result Improved predictive capability and robust long-term statistical performance.
A new autoencoder combines deep learning with SVD to reduce model complexity.
problem Overcoming the Kolmogorov barrier in high-dimensional systems.
method Learnable weighted hybrid autoencoder combining SVD and deep learning.
result Empirically, the model exhibits a sharpness thousands of times smaller than other models.
New EiV models correct bias in operator learning with noisy data.
problem Bias in operator learning due to noisy independent variables.
method Developed EiV models for MOR-Physics and DeepONet.
result EiV models reduce bias in noisy operator learning.
Enhanced ensemble filters use machine learning to improve accuracy in filtering models.
problem Accuracy limitations of traditional ensemble Kalman filters.
method Introduces a measure neural mapping (MNM) to map joint predicted state and observation to updated state estimates.
result Superior root-mean-square-error performance compared to leading methods in filtering models.
Study Galois groupoids of discret Painlevé equations.
problem Computing Galois groupoids for discret Painlevé equations.
method Using semi-continuity theorem for Galois groupoid in confluence of difference to differential equations.
result Computed Galois groupoids for discret Painlevé equations.
Proves solvability of general inverse σ_k equations with constant coefficients.
problem Solvability of general inverse σ_k equations with constant coefficients.
method Proves existence of unique solution if a C-subsolution exists.
result Confirms analytical conjecture for deformed Hermitian--Yang--Mills equation.
We present an unsupervised approach for discovering semantic representations of mathematical equations. Equations are challenging to analyze because each is unique, or nearly unique. Our method, which we call equation embeddings, finds good representations of equations by using the representations of their surrounding …
Paper establishes estimates for nonlinear equations on compact manifolds.
problem Estimating solutions to fully nonlinear equations with gradient terms on compact almost Hermitian manifolds.
method Establishes second order estimates and proves existence of solutions for specific equations.
result Proves existence of solutions for various equations, including Monge-Ampère and Hessian equations.
Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.
problem Elliptic equations on hypercomplex manifolds.
method Proves C^2,alpha estimates under suitable assumptions.
result Solutions to specific elliptic equations on hyperkähler manifolds satisfy C^2,alpha estimates.
The paper generalizes Monge-Ampère equations and their solutions in differential geometry.
problem Understanding the structure of Monge-Ampère equations and their solutions.
method Generalizing Monge-Ampère equations to higher-order systems and proving their solutions correspond to integral manifolds of exterior differential systems.
result The Korteweg-de Vries (KdV) equation and Cauchy-Riemann equations are examples of generalized Monge-Ampère equations.
We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The ot…
Introduces a new PDE involving differential forms for Kähler geometry.
problem Solving a unified PDE for various important equations in Kähler geometry.
method Introduces a fully nonlinear PDE with differential form Λ and proves solvability conditions.
result Generalizes previous works and proves a conjecture for the dHYM equation.
Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.
problem Bounding weak subsolutions of Trudinger's equation on Riemannian manifolds.
method Proving sub-Gaussian upper bounds for weak subsolutions.
result The upper bounds are sharp for specific classes of manifolds, including \(\mathbb{R}^{n}\).
In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in S1,} \end{equation} where (−Δ)21 stands for the fractional Laplacian and κ is a bounded function. We interpret the above equation as the prescri…
The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
problem Gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
method Derives Li-Yau and Souplet-Zhang type gradient estimates for the given equations.
result Gradient estimates for the equations on complete noncompact metric measure spaces with compact boundary.
Paper solves Hessian equations on Kähler manifolds.
problem Solving Hessian equations on Kähler manifolds.
method Combines elementary symmetric functions; provides sufficient and necessary condition.
result Generalizes results for Hessian and Hessian quotient equations.
The paper introduces new equations in Kähler geometry and proves their solutions and convexity.
problem Solving equations in Kähler geometry and understanding their geometric implications.
method Using moment map pictures to motivate and prove solutions for the equations.
result The Mabuchi functional for certain equations is shown to be convex.
We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equatio…
Probabilistic grammars improve equation discovery from data.
problem Discovering scientific laws from data using equations.
method Proposed probabilistic context-free grammars to encode soft constraints and a Monte-Carlo algorithm.
result Probabilistic grammars lead to more efficient equation discovery.
In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equa…
The paper proves constant rank theorems for special Lagrangian equations.
problem Understanding saddle solutions and Liouville type results for special Lagrangian equations.
method Argument based on saddle solutions and Liouville type results for the special Lagrangian equation.
result Obtained constant rank theorems for saddle solutions to the special Lagrangian equation and the quadratic Hessian equation.
Study solves HJB equations for time-inconsistent control problems.
problem Time-inconsistent deterministic linear quadratic control problems.
method Characterized solutions using Riccati equations with integral terms, proving uniqueness.
result Uniqueness of solutions to equilibrium HJB equations proved.
We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialise to various important classes of equ…