We report analytical results for the development of the viscous fingering instability in a cylindrical Hele-Shaw cell of radius a and thickness b. We derive a generalized version of Darcy's law in such cylindrical background, and find it recovers the usual Darcy's law for flow in flat, rectangular cells, with correctio…
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48 results for “Darcy's law”
A new model reduces the cost of simulating fluid flow through porous materials.
problem High computational cost of simulating fluid flow through porous materials.
method Proposes a fully probabilistic, Darcy-type reduced-order model.
result The model significantly accelerates uncertainty quantification tasks.
Over the past few years, we developed a mathematically rigorous method to study the dynamical processes associated to nonlinear Forchheimer flows for slightly compressible fluids. We have proved the existence of a geometric transformation which relates constant mean curvature surfaces and time-invariant pressure distri…
Sequential Bayesian Design for Efficient Surrogate Construction in the Inversion of Darcy Flowsstat.ML
This study proposes an efficient surrogate for Darcy flow inverse problems.
problem Efficiently constructing accurate surrogate models for high-dimensional complex inverse problems.
method Sequential Bayesian design strategy to acquire a locally accurate surrogate model focusing on high-probability regions.
result The proposed method accelerates inversion accuracy and computational speed.
Developed MF-PINNs to solve coupled Stokes-Darcy equations more accurately.
problem Solving coupled Stokes-Darcy equations with varying physical constants.
method Combining VP and SV forms with adjusted weights in MF-PINNs.
result Improved accuracy of streamline and pressure fields in numerical experiments.
We derive a numerical method for Darcy flow, hence also for Poisson's equation in mixed (first order) form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is one of its discretizations on simplicial complexes such as triangle and tetrahedr…
This work represents an application of constant mean curvature graphs (as solutions of the mean curvature PDE) to non-linear non-Darcy flows in porous media. It relates time invariant pressure distribution graphs to graphs of constant mean curvature surfaces. This differential geometric interpretation provides an impor…
A new training method improves stability and generalization of DeepONets.
problem Training deep operator networks (DeepONets) is challenging due to nonconvex and nonlinear nature.
method Two-step training method: first train trunk network, then branch network. Introduced Gram-Schmidt orthonormalization.
result Generalization error estimate and numerical examples demonstrating effectiveness.
Adaptive operator learning reduces costs in Bayesian inverse problems.
problem Reducing computational costs in Bayesian inverse problems governed by PDEs.
method Adaptive operator learning framework that gradually reduces modeling error.
result The approach significantly reduces computational costs while maintaining inversion accuracy.
Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We s…
A natural generalization of a crossing change is a rational subtangle replacement (RSR). We characterize the fundamental situation of the rational tangles obtained from a given rational tangle via RSR, building on work of Berge and Gabai, and determine the sites where these RSR may occur. In addition we also determine …
PILNO uses neural operators to solve PDEs efficiently on point clouds.
problem Solving partial differential equations (PDEs) on point cloud data efficiently.
method Physics-informed low-rank neural operator framework combining low-rank kernel approximations and an encoder-decoder architecture.
result PILNO efficiently approximates solution operators of PDEs on point cloud data, satisfying PDE constraints and boundary conditions.
This work explains scaling laws as redundancy laws in deep learning.
problem The mathematical origins of scaling laws in deep learning models remain unclear.
method Kernel regression and analysis of data covariance spectra.
result Scaling laws can be explained as redundancy laws, revealing the learning curve's slope depends on data redundancy.
Study on RL on volatility surfaces, proving no free lunch for law-seeking methods.
problem Aligning RL agents with no-arbitrage laws in volatile markets.
method Built a law manifold, defined penalties, and used a Goodhart decomposition.
result No free lunch theorem: Law-seeking RL cannot outperform baselines.
Machine learning identifies math sequences based on empirical laws.
problem Identifying interesting mathematical structures.
method Extract features from integer sequences using Benford's and Taylor's laws; experiment with classifiers.
result Machine learning can identify various mathematical properties in sequences.
Visualizes Bangladeshi laws for quicker searching.
problem Difficulty in finding relevant Bangladeshi laws.
method Doc2Vec for node layout, link mining for citation networks, named entity recognition for quick section finding.
result Users find the tool faster and more intuitive for legal research.
A new scaling law predicts optimal batch size for training models.
problem Finding the optimal batch size for training models efficiently.
method Proposed a three-term scaling law that considers model size, training data, training steps, and batch size.
result The three-term law accurately recovers the optimal batch size and can be robustly fit with fewer training runs.
Space exploration technology advances exponentially, consistent with Moore's and Wright's laws.
problem Predicting the advancement of space exploration technology.
method Analysis of Moore's and Wright's laws applied to space exploration technology.
result Spacecraft technology advances exponentially, consistent with Moore's and Wright's laws.
This work analyzes neural scaling laws using power-law data spectra and derives analytical expressions for generalization error.
problem Understanding how neural network performance scales with key factors like data size and model complexity.
method Statistical mechanics techniques applied to one-pass stochastic gradient descent in a student-teacher framework.
result Derivation of analytical expressions for generalization error under power-law data spectra and identification of conditions for power-law scaling.
Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Lawsmath.AP
Study finds conservation laws for a specific class of parabolic equations.
problem Existence and structure of conservation laws for evolutionary scalar second-order differential equations.
method Calculation of linearized characteristic cohomology to find conservation laws, showing dependence on second derivatives.
result Only Monge-Ampère type equations have non-trivial conservation laws.
An Atlas model is a rank-based system of continuous semimartingales for which the steady-state values of the processes follow a power law, or Pareto distribution. For a power law, the log-log plot of these steady-state values versus rank is a straight line. Zipf's law is a power law for which the slope of this line is …
Large models follow power laws in performance with dataset size or parameters.
problem Understanding neural scaling laws in large language models.
method Joint generative data model and random feature model.
result Modeling and solving the dual limit reveals insights into scaling laws.
Conservation law for weakly harmonic mappings in high dimensions.
problem Conservation law for harmonic mappings in supercritical dimensions.
method Partial extension of Rivière's conservation law with Lorentz integrability condition.
result Conservation law for weakly harmonic mappings in supercritical dimensions.
NOGaP uses neural operators and GPs to solve PDEs with uncertainty quantification.
problem Lack of uncertainty measures in neural operator solutions for PDEs.
method NOGaP combines neural operators with Gaussian Processes to provide probabilistic solutions.
result NOGaP offers improved prediction accuracy and uncertainty quantification.
This paper tackles regularization parameter learning in inverse problems using data-driven bilevel optimization.
problem Finding optimal regularization parameters in inverse problems.
method Data-driven bilevel optimization approach, analyzing performance in large data samples.
result The approach can reduce computational cost through online numerical schemes based on stochastic gradient descent.
Survey on conservation laws for geometric PDEs.
problem Modeling polyharmonic maps.
method Conservation law approach.
result Overview of conservation laws in geometric PDEs.
Dynamic risk measures follow law invariance principles over time.
problem Tackles dynamic risk measurement principles.
method Shows equivalence between adapted law invariance and recursive one-step conditional-law representation for time-consistent risk measures.
result Identifies adapted law invariance as the dynamic counterpart of ordinary law invariance.
Learning curves theory for hierarchically compositional data with power-law distributed featuresstat.ML
Unified theory for neural scaling laws in hierarchically compositional data.
problem Understanding neural scaling laws in hierarchically compositional data.
method Probabilistic context-free grammars and power-law distributed production rules.
result Unified learning curve behavior for classification and next-token prediction tasks.
Defines formal vertex laws related to Lie conformal algebras.
problem No specific problem stated; focuses on definitions and proofs.
method Definitions and proofs of vertex/conformal versions of classical Lie theory results.
result Proves vertex/conformal versions of important Lie theory results.
Theory of Zipf's Law and of General Power Law Distributions with Gibrat's law of Proportional Growthq-fin.GN
We summarize a book under publication with his title written by the three present authors, on the theory of Zipf's law, and more generally of power laws, driven by the mechanism of proportional growth. The preprint is available upon request from the authors. For clarity, consistence of language and conciseness, we disc…
New concept of partial law invariance connects decision theory and financial risk management.
problem Connecting decision theory and financial risk management under uncertainty.
method Characterizing partially law-invariant coherent risk measures via a novel representation formula.
result Strong partial law invariance bridges the gap between existing risk measure representations.
This work extends the scaling law to multiple and kernel regression, challenging traditional machine learning principles.
problem Challenging traditional machine learning wisdom with scaling law in large practical models.
method Demonstrates the scaling law in multiple and kernel regression settings.
result The scaling law extends to multiple and kernel regression, providing deeper insights into LLMs.
Deformation of semi-circle law for the correlated time series and Phase transitioncond-mat.stat-mech
Study on eigenvalue distribution of correlated time series deforming the semi-circle law.
problem Eigenvalue distribution of correlated time series differs from the semi-circle law.
method Analysis of Wigner random matrix with temporal correlation.
result Eigenvalue distribution converges to a deformed semi-circle law with longer tail and higher peak.
The article discusses conservation laws for polyharmonic maps and their applications.
problem Understanding conservation laws for polyharmonic maps.
method Recalling the stress-energy tensor and showing conservation laws with Killing vector fields.
result Conservation laws for polyharmonic maps and their applications.
Market lawsq-fin.ST
Power laws detected in financial data, modeled with random multipliers.
problem Detecting power laws in financial data.
method Investigated data from financial instruments, proposed a model based on sums of Maxwell-Boltzmann distributions with random multipliers.
result Detected power laws with various exponents in financial data, proposed a universal model.
Non-Gibrat's law in the middle scale regionphysics.soc-ph
By using numerical simulation, we confirm that Takayasu--Sato--Takayasu (TST) model which leads Pareto's law satisfies the detailed balance under Gibrat's law. In the simulation, we take an exponential tent-shaped function as the growth rate distribution. We also numerically confirm the reflection law equivalent to the…
Empirical law predicts accuracy of Google Translate's translation chains.
problem Predicting accuracy in machine translation with multiple hops.
method Empirical testing of Google Translate's sequential translation.
result Accuracy decreases with the number of translating hops, following a power law.
Machine learning finds new natural laws from noisy data.
problem Discovery of natural laws relies on human inspiration.
method Recursive-LASSO-based symbolic regression (RLS) method.
result Data-driven formulation of natural laws from noisy data.
Extends results for law-invariant functionals to random variable spaces.
problem Establishing results for a broad class of random variable spaces.
method Using structural results for law-invariant functionals and extending to new spaces.
result Unified perspective on law-invariant functionals, including quantile-based representations.
The distribution of share prices follows Zipf's law, similar to financial indicators.
problem Investigating the distribution of share prices and financial indicators.
method Statistical analysis of data from 8,000 companies over 10 years.
result Share price and financial indicators follow Zipf's law, indicating that corporate value follows Zipf's law.
Employing profits data of Japanese companies in 2002 and 2003, we identify the non-Gibrat's law which holds in the middle profits region. From the law of detailed balance in all regions, Gibrat's law in the high region and the non-Gibrat's law in the middle region, we kinematically derive the profits distribution funct…
New SDE model from machine learning optimization with unique stationary distribution.
problem Stationary distribution of machine learning optimization models.
method Proved ergodicity and unique stationary distribution of power-law dynamic SDE.
result Power-law dynamic has a unique stationary distribution and is ergodic.
Study reveals neural scaling laws in random graphs and natural language models.
problem Understanding the origin of neural scaling laws in complex systems.
method Examined scaling laws in transformers trained on random walks and simplified natural language models.
result Neural scaling laws emerge in the absence of power law structure in data correlations.
Researchers found a geometric duality for systems of conservation laws.
problem Systems of conservation laws and their additional conservation laws.
method Assigning ruled surfaces in projective space and defining dual systems.
result Hamiltonian systems are autodual, and 3-component nondiagonalizable systems are dual to systems with constant characteristic speeds.
Proves Weyl's law for metric spaces with Ricci curvature.
problem Proving Weyl's law for metric measure spaces with bounded Ricci curvature.
method Analyzes spaces to prove asymptotic eigenvalue formula.
result Establishes Weyl's law for Dirichlet eigenvalues in metric measure spaces.
Study finds conservation laws for maps on spheres.
problem Conservation laws for maps on spheres.
method Derives conservation laws for Dirac-harmonic maps and their extensions.
result Conservation laws for maps on spheres.
New framework reveals thermodynamic principles for LLM training.
problem Understanding the training dynamics of large language models.
method Introducing Neural Thermodynamic Laws (NTL) under river-valley loss landscape assumptions.
result Key thermodynamic quantities and principles naturally emerge in LLM training.
Develops a simple model to understand learning curves for arbitrary power laws.
problem Lack of theoretical understanding of scaling laws in machine learning.
method Analyzes a toy model to determine if learning curves are universal or depend on data distribution.
result Determines that learning curves can exhibit for arbitrary power .