Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial differential equations, are presented, which arise naturally in some longstanding, …
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48 results for “differential techniques”
These notes introduce key techniques in differential geometry for curves and surfaces.
problem Understanding the basics of differential geometry for curve and surface analysis.
method Problem-centered, elementary, visual approach to teaching essential techniques.
result Provides a solid foundation for further study in differential geometry.
Study evaluates federated learning with differential privacy on MIMIC-III, improving model performance with careful parameter tuning.
problem Training machine learning models on privacy-sensitive data sets locked in healthcare facilities.
method Extensive evaluation of federated and differential privacy techniques on MIMIC-III dataset, analyzing various parameters.
result Careful parameter tuning is crucial for federated learning with differential privacy, especially for data distribution and communication strategies.
Derivatives, mostly in the form of gradients and Hessians, are ubiquitous in machine learning. Automatic differentiation (AD), also called algorithmic differentiation or simply "autodiff", is a family of techniques similar to but more general than backpropagation for efficiently and accurately evaluating derivatives of…
Study uses graph techniques to understand meromorphic quadratic differential strata.
problem Understanding the topology of meromorphic quadratic differential strata.
method Exchange graph techniques to study fundamental groups; generalizes relations for mixed-angulations.
result Explicit presentations of fundamental groups in genus-zero case with four singularities.
Broad adoption of machine learning techniques has increased privacy concerns for models trained on sensitive data such as medical records. Existing techniques for training differentially private (DP) models give rigorous privacy guarantees, but applying these techniques to neural networks can severely degrade model per…
Researchers develop methods to calibrate ABMs using Bayesian techniques.
problem Challenges in calibrating ABMs due to likelihood intractability and non-differentiability.
method Generalised variational inference for differentiable ABMs.
result Accurate Bayesian parameter inferences for differentiable ABMs demonstrated.
The paper extends statistical estimation techniques under differential privacy.
problem Establishing sample complexity bounds for estimation tasks under differential privacy.
method Proposes analogues of Le Cam's method, Fano's inequality, and Assouad's lemma under central differential privacy.
result Optimal sample complexity bounds for discrete distribution estimation under total variation and distances.
New proof of Gaffney's inequality for differential forms on manifolds with boundary.
problem Proving Gaffney's inequality for differential forms on manifolds with boundary.
method Variational approach combined with Bochner's technique.
result New proof of Gaffney's inequality for differential forms.
Paper proposes a black-box technique to generate adversarial samples.
problem Robustness of Deep Neural Networks (DNNs) to adversarial samples.
method Black-box Momentum Iterative Fast Gradient Sign Method (BMI-FGSM) using Differential Evolution to approximate gradients.
result Achieves high success rates in generating adversarial samples and misclassification.
DAEGEN generates adversarial inputs for neural networks using a black-box differential technique.
problem Generating adversarial inputs that highlight differences between neural network models.
method DAEGEN uses a local search-based optimization algorithm to find difference-inducing adversarial examples (DIAEs).
result DAEGEN is the first black-box differential technique for adversarial input generation and performs well compared to existing methods.
Local Kan conditions enable differentiation of simplicial manifolds.
problem Differentiating simplicial manifolds into Lie algebroids.
method Expanding a technique for higher Lie groupoids to simplicial manifolds.
result Derivation of a method to differentiate simplicial manifolds into higher Lie algebroids.
Explains curves and surfaces in differential geometry.
problem Understanding smooth curves and surfaces in differential geometry.
method Problem-centered, elementary, visual approach focusing on essential techniques.
result Provides a solid foundation for further study in differential geometry.
We characterize primary operations in differential cohomology via stacks, and illustrate by differentially refining Steenrod squares and Steenrod powers explicitly. This requires a delicate interplay between integral, rational, and mod p cohomology, as well as cohomology with U(1) coefficients and differential forms. A…
Develops experimental design for discovering missing physics in bioreactors.
problem Discovering missing physics in incomplete model structures of process systems.
method Combines universal differential equations and symbolic regression with sequential experimental design.
result Successfully recovered true model structure of a bioreactor using machine learning techniques.
These are lecture notes of the Summer school on the geometry of differential equations held in Nordfjordeid, Norway in 1996. They cover geometric structures related to scalar second order ODEs, the construction of the associated Cartan connection, techniques for computing invariants of differential equations starting f…
Various stochastic models have been proposed to estimate mortality rates. In this paper we illustrate how machine learning techniques allow us to analyze the quality of such mortality models. In addition, we present how these techniques can be used for differentiating the different causes of death in mortality modeling…
We review and elaborate on some aspects of the quantization of certain classes of higher abelian gauge theories using techniques of generalized differential cohomology. Particular emphasis is placed on the examples of generalized Maxwell theory and Cheeger-Simons cohomology, and of Ramond-Ramond fields in Type II super…
Confirming a conjecture, we show fundamental groups of certain abelian differentials are framed mapping class groups.
problem Confirming a 1997 conjecture about fundamental groups of abelian differentials.
method Algebraic-geometric approach using Shimada's techniques for computing fundamental groups via morphisms of varieties.
result Orbifold fundamental groups of these strata are described as framed mapping class groups.
Gradient estimation techniques applied to programs with randomness in high energy physics.
problem Differentiating programs with discrete randomness in high energy physics.
method Several gradient estimation techniques, including Stochastic AD method, applied to simplified detector design experiments.
result Development of the first fully differentiable branching program.
We study -divergence contraction and its privacy implications.
problem Analyzing privacy in data processing and algorithms.
method Generalizing Dobrushin's coefficient to -divergence and deriving contraction coefficients.
result Local differential privacy can be expressed in terms of -divergence contraction, leading to precise sample size reductions.
We study `constrained generalized Killing (s)pinors', which characterize supersymmetric flux compactifications of supergravity theories. Using geometric algebra techniques, we give conceptually clear and computationally effective methods for translating supersymmetry conditions into differential and algebraic constrain…
Stochastic optimization techniques are standard in variational inference algorithms. These methods estimate gradients by approximating expectations with independent Monte Carlo samples. In this paper, we explore a technique that uses correlated, but more representative , samples to reduce estimator variance. Specifical…
Global homotopies upgrade classical map in differential geometry.
problem Upgrade classical Hochschild-Kostant-Rosenberg map to a deformation retract.
method Combining symbol calculus and coalgebraic van Est theorem.
result Develop deformation retracts in various settings.
This paper extends AD techniques to Monte Carlo processes for efficient derivative calculation.
problem Obtaining derivatives of expectation values in Monte Carlo processes.
method Two approaches: reweighting and Hamiltonian extension of HMC.
result Hamiltonian approach as a change of variables simplifies variance reduction.
New bounds for private learning of high-dimensional Gaussian distributions.
problem Learning high-dimensional Gaussian distributions under differential privacy constraints.
method Analytic tools for constructing global covers from local covers, modified hypothesis selection techniques.
result Near-optimal sample complexity bounds for general Gaussians, conjectured to be near-optimal in the general case.
We prove that the 2-primary is zero. As a consequence, the Kervaire invariant element is contained in the strictly defined 4-fold Toda bracket . Our result has a geometric corollary: the 61-sphere has a unique smooth structure and it is the last odd dimensional case - the o…
Using stable log maps, we introduce log twisted differentials extending the notion of abelian differentials to the Deligne-Mumford boundary of stable curves. The moduli stack of log twisted differentials provides a compactification of the strata of abelian differentials. The open strata can have up to three connected c…
Framework purifies approximate differential privacy to pure differential privacy.
problem Achieving pure differential privacy from approximate differential privacy.
method Randomized post-processing with calibrated noise to eliminate δ parameter.
result First statistically and computationally efficient reduction from approximate DP to pure DP.
New framework improves differential privacy for asymmetric datasets.
problem Improving differential privacy for asymmetric datasets.
method Adapts inverse sensitivity mechanism with sparse vector technique.
result Efficiently estimates general functions with improved privacy.
Efficiently estimates quantiles and maximum in unbounded datasets with differential privacy.
problem Efficiently estimating quantiles and maximum in unbounded datasets with differential privacy.
method Simple invocation of a subroutine called AboveThreshold, iteratively called in Sparse Vector Technique.
result Improved estimates on highest quantiles with robustness and accuracy.
BUDS balances privacy and utility by shuffling data, achieving strong privacy with minimal loss.
problem Balancing privacy and utility in crowd-sourced statistical databases.
method One-hot encoding, iterative shuffling, loss estimation, risk minimization.
result Achieves for privacy, maintaining a privacy bound of .
Develops a computationally tractable high-dimensional differential privacy estimator.
problem Differential privacy in high dimensions is computationally intractable.
method Combines high-dimensional robust statistics with differential privacy techniques.
result A computationally tractable algorithm with dimension-independent privacy loss.
Classifies scalar second-order PDEs with low-dimensional symmetry groups.
problem Classifying differential equations with specific symmetry groups.
method Algebraic technique based on covariant form for constructing equations.
result Complete classification of quasi-linear scalar second-order PDEs with free symmetry groups of dimension ≤3.
Survey on symmetry in manifold structures.
problem Classifying manifolds with differential-geometric structures.
method Algebra, dynamics, and analysis techniques.
result Illustration of various techniques in action.
In this paper, we introduce a large class of convergent numerical methods, based on (linear) basis function regression technique, to approximate the solution to a forward-backward stochastic differential equation with jumps (FBSDEJ hereafter). Numerical experiment shows good applicability of the proposed method.
Modern differential cohomology explained with applications.
problem Understanding differential cohomology from a modern perspective.
method Sheaves on manifolds, Chern-Weil theory, differential characters, differential characteristic classes.
result Differential lift of the first Pontryagin class.
Automatic differentiation---the mechanical transformation of numeric computer programs to calculate derivatives efficiently and accurately---dates to the origin of the computer age. Reverse mode automatic differentiation both antedates and generalizes the method of backwards propagation of errors used in machine learni…
Differentiable methods fail due to spectral issues in Jacobians.
problem Chaos-based failures in differentiable programming.
method Trace failures to the spectrum of Jacobians.
result Criteria for avoiding spectral failure in optimization.
There has been rapid progress recently on the application of deep networks to the solution of partial differential equations, collectively labelled as Physics Informed Neural Networks (PINNs). In this paper, we develop Physics Informed Extreme Learning Machine (PIELM), a rapid version of PINNs which can be applied to s…
New privacy method for eye tracking data reduces correlations and maintains accuracy.
problem Privacy concerns in eye tracking data from VR/AR glasses.
method Transform-coding based differential privacy mechanism for eye movement data.
result Significant reductions in sample correlations and query sensitivities, providing high privacy without loss in accuracy.
SMOTE-DP enhances synthetic data privacy without sacrificing utility.
problem Balancing privacy and utility in synthetic data generation.
method Integrating SMOTE with differential privacy mechanisms.
result SMOTE-DP produces synthetic data that is both private and useful.
As is known, an option price is a solution to a certain partial differential equation (PDE) with terminal conditions (payoff functions). There is a close association between the solution of PDE and the solution of a backward stochastic differential equation (BSDE). We can either solve the PDE to obtain option prices or…
The seminal work of Eskin-Masur-Zorich described the principal boundary of moduli spaces of abelian differentials that parameterizes flat surfaces with a prescribed generic configuration of short parallel saddle connections. In this paper we describe the principal boundary for each configuration in terms of twisted dif…
Machine learning techniques based on neural networks are achieving remarkable results in a wide variety of domains. Often, the training of models requires large, representative datasets, which may be crowdsourced and contain sensitive information. The models should not expose private information in these datasets. Addr…
Paper improves privacy bounds for shuffle model using novel numerical techniques.
problem Improving privacy guarantees in the shuffle model of differential privacy.
method Develops and evaluates numerical techniques for tighter -differential privacy bounds.
result Accurately evaluates privacy loss distribution for adaptive compositions of shufflers.
Variational Optimizationstat.ML
We discuss a general technique that can be used to form a differentiable bound on the optima of non-differentiable or discrete objective functions. We form a unified description of these methods and consider under which circumstances the bound is concave. In particular we consider two concrete applications of the metho…
The paper counts ends of differential forms on surfaces.
problem Counting ends of meromorphic 1-forms on Riemann surfaces.
method Degeneration techniques and moduli space construction.
result Enumeration of ends for meromorphic 1-forms.