Paper discovers governing equations from data using differential invariants.
problem Discovering partial differential equations from data is challenging.
method The paper proposes a pipeline based on differential invariants to reduce the search space and adhere to symmetry.
result DI-SINDy method outperforms other symmetry-informed methods in PDE discovery.
Discover governing equations from data without specifying terms.
problem Discovering differential equations from data without predefined terms.
method Data-driven approach using genetic programming and automatic differentiation.
result Calibrated differential equations from various solutions of a differential equation.
New algorithms recover differential equations from short bursts of data.
problem Locally recover unknown governing differential equations from measurement data.
method Approximate governing equations using standard basis functions and short bursts of trajectory data.
result Effective numerical algorithms recover accurate governing equations from short bursts of data.
Paper formulates governing equations for membrane O surfaces.
problem Formulating equations for membrane O surfaces.
method Formulated governing equations for membrane O surfaces of the 1st and 2nd kind.
result Membrane O surfaces are a subclass of Demoulin's Ω surfaces.
This work discovers governing equations from limited data using physics-informed deep learning.
problem Discovering governing equations from scarce and noisy data for complex systems.
method Physics-informed deep learning framework integrating neural networks, physics embedding, and sparse regression.
result The method effectively identifies governing equations from various spatiotemporal systems with different levels of data scarcity and noise.
Deep neural networks approximate unknown governing equations from data.
problem Approximating unknown governing equations from observational data.
method Residual network (ResNet) and multi-step methods (RT-ResNet, RS-ResNet) for equation recovery.
result Deep neural networks can recover governing equations without time derivative data.
New method identifies physical processes and parameters from data.
problem Current data-driven methods assume known or linear model parameters, limiting realistic process identification.
method Combines data-driven and data-assimilation methods for simultaneous process and parameter identification.
result Successfully identifies physical processes and infers model parameters for nonlinear models.
Koopman Regularization learns governing equations from sparse data.
problem Learning governing equations from sparse and corrupted data.
method Constrained optimization using Koopman Eigenfunctions.
result Restores dynamics precisely with minimal assumptions.
Paper discovers structural dynamics equations from only acceleration data.
problem Discovering equations from only acceleration measurements in structural dynamics.
method Library-based approach with Approximate Bayesian Computation (ABC) prioritizing parsimonious models.
result Efficacy demonstrated in four structural dynamics examples, including linear and nonlinear systems.
Bayesian autoencoders discover physics from noisy data.
problem Challenges in identifying governing equations and coordinates from noisy, low-data real-world data.
method Bayesian SINDy autoencoders with hierarchical Bayesian sparsifying prior and adaptive empirical Bayesian method.
result Better physics discovery with lower data and fewer training epochs, along with valid uncertainty quantification.
New methods distill dynamical system equations from data.
problem Discovering governing equations of dynamical systems from data.
method Sparse regression, LASSO, dual LASSO optimization, STRidge algorithm.
result Improved accuracy and stability in learning dynamical system equations.
NKN deep neural network learns governing equations and classifies images.
problem Learning governing equations and classifying images with deep neural networks.
method Nonlocal kernel network (NKN) that is resolution independent, deep, and handles various tasks.
result NKN outperforms baseline methods in learning governing equations and image classification tasks.
Deformations of curves on Minkowski plane governed by mKdV equation.
problem Deformations of curves on the Minkowski plane.
method Isoperimetric deformation governed by the defocusing mKdV equation.
result Explicit formula for curve motion on Minkowski plane, including singular points.
New method extracts stochastic laws from data, including Lévy noise.
problem Extracting stochastic laws from data with non-Gaussian noise.
method Using normalizing flows to estimate transition density, then applying nonlocal Kramers-Moyal formulas.
result Can learn stochastic differential equations with Lévy motion.
New techniques improve the accuracy of identifying nonlinear systems from noisy data.
problem Identifying nonlinear dynamical systems from noisy state measurements.
method Comparative study of local and global smoothing techniques to denoise state measurements and improve sparse regression methods.
result Global smoothing methods outperform local methods in improving the accuracy of governing equation recovery.
Proposes a framework to identify and correct model-form errors in nonlinear systems.
problem Model-form errors in nonlinear dynamical systems due to unknown or approximated governing equations.
method Uses a hybrid approach combining machine learning and Bayesian filtering to estimate and correct model-form errors.
result Improves the predictive capability of known but approximate governing equations for nonlinear dynamical systems.
Stock market returns follow q-Gaussian distributions with super-diffusion.
problem Characterizing stock market price returns.
method Used q-Gaussian distributions and porous media equation to model stock market returns.
result Stock market returns follow q-Gaussian distributions with super-diffusion.
We show that the degenerate special Lagrangian equation, recently introduced by Rubinstein-Solomon, induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, …
Machine learning identified 13 key equations for distillation column dynamics.
problem Identify governing laws for complex engineered systems.
method Sparse Identification of Non-Linear Dynamics (SINDy) applied to distillation column data.
result Reduced 1000s of equations to 13 interpretable terms.
Affine manifolds linked to integrable equations and geometric structures.
problem Understanding the geometric and algebraic properties of affine manifolds.
method Analyzing the Kahlerian tangent bundle and multi-dimensional consistency of the TED equation.
result Affine manifolds are related to self-dual Einstein spaces and Hessian structures.
Develops a new method to discover stochastic systems with non-Gaussian noise.
problem Discovering governing laws from complex systems with non-Gaussian noise.
method Theoretical framework and numerical algorithm to extract stochastic differential equations with Gaussian and non-Gaussian noise.
result Demonstrated the efficacy and accuracy of the approach on various systems.
We propose a new cognitive framework for option price modelling, using quantum neural computation formalism. Briefly, when we apply a classical nonlinear neural-network learning to a linear quantum Schrödinger equation, as a result we get a nonlinear Schrödinger equation (NLS), performing as a quantum stochastic filter…
SIP framework discovers governing equations in uncertain systems.
problem Discovering governing equations in systems with input variability and noisy data.
method SIP framework treats unknown coefficients as random variables and infers their posterior distribution by minimizing Kullback-Leibler divergence.
result SIP consistently identifies correct equations and lowers coefficient error by 82% relative to SINDy.
Deep neural network generates symbolic equations from data.
problem Lack of insight into underlying mappings from traditional deep learning.
method Combines deep learning flexibility with symbolic solutions.
result Accurately generates governing equations for dynamical systems.
It is demonstrated that hypersurfaces with a flat centroaffine metric are governed by a system of nonlinear PDEs known as the equations of associativity of 2-dimensional topological field theory.
A universal rule-based self-learning approach using deep reinforcement learning (DRL) is proposed for the first time to solve nonlinear ordinary differential equations and partial differential equations. The solver consists of a deep neural network-structured actor that outputs candidate solutions, and a critic derived…
In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an sl(2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary d…
Method extracts governing laws from non-Gaussian stochastic systems data.
problem Modeling complex dynamics with non-Gaussian Lévy noise.
method Data-driven method to extract stochastic dynamical systems from noisy data.
result Established a theoretical framework and numerical algorithm to compute Lévy jump measure, drift, and diffusion.
Paper solves stock loan pricing with finite maturity using integral equations.
problem Valuation of margin-call stock loans with finite maturities.
method Fourier Sine transform and Volterra integral equation approach.
result Integral representation of margin-call stock loan value.
Method improves SINDy for noisy nonlinear systems.
problem Recover nonlinear dynamical systems from noisy data.
method Reweighted ℓ1-regularized least squares. result Improved accuracy and robustness in noisy conditions.
Scroll structures on solutions of 4D integrable equations are involutive and governed by a dispersionless hierarchy.
problem Characterizing the geometry of solutions to 4D integrable equations.
method Defining rational normal scrolls and showing their involutivity.
result Involutive scroll structures are governed by a dispersionless integrable hierarchy.
Paper presents MF-PIDNN for physics-informed deep learning with low-fidelity data.
problem Challenges in systems with unknown or approximate governing differential equations and limited high-fidelity data.
method Transfer learning between physics-informed and data-driven deep learning models.
result Model provides accurate predictions even in data-scarce regions.
Machine learning identifies physical laws from data, but discrepancies remain.
problem Discovering universal physical laws from data alone is challenging.
method Sparse Identification of Nonlinear Dynamics (SINDy) method to identify governing equations.
result Measurement noise and secondary physical mechanisms obscure the underlying law of gravitation.
Derives neural network equations from physics and math, clarifying their geometry.
problem Understanding the geometry and mechanics of neural networks.
method Derives neural network equations using Fermat's principle, exterior differential forms, and differential geometry.
result Shows how the loss function acts like a Hamiltonian and how the layer metric explains pretraining.
Deep learning scheme identifies and reconstructs chaotic and stochastic systems from noisy data.
problem Challenging identification of governing equations from noisy and partial observations.
method Jointly learns inference model and governing laws using variational deep learning.
result Framework generalizes state-of-the-art methods and accounts for stochastic variabilities.
We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initial submanifolds. Each such submanifold corresponds to an orbit of the holonomy group on the modelling homogeneous space and car…
Continuous time random walks impose a random waiting time before each particle jump. Scaling limits of heavy tailed continuous time random walks are governed by fractional evolution equations. Space-fractional derivatives describe heavy tailed jumps, and the time-fractional version codes heavy tailed waiting times. Thi…
NeuPDE uses neural networks to model time-dependent data using differential equations.
problem Modeling time-dependent data from dynamic datasets.
method Neural network approach with both shallow multilayer perceptrons and nonlinear differential terms.
result Demonstrated on various dynamical systems, NeuPDE outperforms other methods.
A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of functions obeying the classical linear Darboux equations for conjugate nets.The distinguished role of the Euler-Poisson-Darboux equations and associated Lauricella-type functions is em…
Model analyzes systemic risk in banking systems using stochastic differential equations.
problem Govern systemic risk in banking systems.
method Stochastic differential equations, optimal control problem, pseudo mean field approximation.
result Monetary authority can control systemic risk by optimizing bank behavior.
Deep neural networks with memory learn reduced equations from partial data.
problem Constructing governing equations for unknown dynamical systems from limited data.
method Formulate a discrete approximation of memory integrals, use deep neural networks to incorporate history terms.
result Deep neural networks can learn reduced equations with memory from partial data.
This review reports some theoretical results on the Geometry of membranes. The governing equations to describe equilibrium configurations of lipid vesicles, lipid membranes with free edges, and chiral lipid membranes are derived from the variation of free energies of these structures. Some analytic solutions to these e…
AutoKE automates embedding physical knowledge into neural networks for complex engineering problems.
problem Complex physical equations in engineering problems.
method AutoKE framework using deep neural networks, equation parsing, automatic differentiation, adaptive weights, and NAS.
result Automatically embeds physical knowledge into neural networks for complex equations efficiently.
A new algorithm speeds up sparse regression for discovering equations from data.
problem Learning governing equations from vast data with unsatisfying descriptions.
method SPRINT: a fast algorithm using bisection and analytic bounds to identify optimal rank-1 modifications.
result A calculation that would take millions of years can be done in a day.
New method learns low-dimensional models for systems with non-polynomial terms.
problem Modeling systems with non-polynomial nonlinear terms that are spatially local and given in analytic form.
method Non-intrusive model reduction method that learns operators for linear and polynomially nonlinear dynamics via a least-squares problem incorporating given non-polynomial terms.
result Comparable accuracy to intrusive methods that require full knowledge of governing equations.
DL-PDE discovers PDEs from noisy, sparse data using neural networks and sparse regressions.
problem Discovering PDEs from noisy, sparse data.
method Combines neural networks and sparse regressions to discover PDEs from meta-data generated by a neural network.
result Achieves satisfactory results in real-world engineering settings with noisy and limited data.
Automates discovery of interpretable Lagrangians from data.
problem Lack of interpretable Lagrangian discovery methods.
method Data-driven machine learning algorithm to derive interpretable Lagrangians.
result Automated discovery of interpretable Lagrangians and conservation laws.
Recent theoretical advances in elasticity of membranes following Helfrich's famous spontaneous curvature model are summarized in this review. The governing equations describing equilibrium configurations of lipid vesicles, lipid membranes with free edges, and chiral lipid membranes are presented. Several analytic solut…