Neural networks can approximate complex stochastic equations well.
problem Approximating general stochastic differential equations.
method Identified neural network classes approximating continuous functions.
result Neural stochastic differential equations can approximate general stochastic differential equations arbitrarily well.
Pruning neural networks adds differential privacy noise, preserving data utility.
problem Achieving differential privacy in neural networks without sacrificing data utility.
method Proving equivalence between pruning and adding differential privacy noise to hidden-layer activations.
result Pruning can be a more effective alternative to adding differential privacy noise for neural networks.
DiffEqFlux.jl integrates neural networks with differential equations.
problem Combining machine learning and differential equations for modeling complex systems.
method Fusing neural networks and differential equations using DiffEqFlux.jl.
result Demonstrates the integration of differential equations into neural networks and vice versa.
This work integrates differentiation and integration in Physics-Informed Neural Networks.
problem Solving integro-differential equations and computing integral transforms.
method Augmenting Physics-Informed Neural Networks with automatic integration.
result Solving complex integral transforms and integro-differential equations.
Efficient neural networks compute various differential operators cheaply.
problem Efficient computation of higher time complexity differential operators.
method Restricted neural network architectures with diagonal and hollow Jacobian matrices, allowing efficient extraction of dimension-wise derivatives.
result Demonstrated efficient computation of differential operators for various applications.
New neural network uses differential equations for adaptable activation functions.
problem Fixed activation functions limit neural network performance and size.
method Introduces differential equation units (DEUs) that learn nonlinear activation functions.
result DEUs enable more compact networks with comparable performance.
Differential equations explain neural network dynamics.
problem Understanding neural networks using differential equations.
method Analyzing properties of differential equations as machine learning models.
result Neural networks can be modeled by differential equations.
Physics-informed neural networks solve PDEs using neural networks.
problem Solving nonlinear partial differential equations (PDEs) with neural networks.
method Physics-informed neural networks trained to solve PDEs while respecting physical laws.
result Physics-informed neural networks can infer solutions to PDEs and create differentiable surrogate models.
New framework explains neural network bias in solving differential equations.
problem Understanding and controlling the bias in PINNs for differential equations.
method Deriving an integro-differential equation from PINNs and GPR equivalence.
result PINN predictions are influenced by a kernel term reflecting architecture choices.
Elvet solves differential equations and variational problems with neural networks.
problem Solving complex differential and variational equations with arbitrary conditions.
method Machine learning, specifically neural networks, to represent and solve equations.
result Elvet can solve a wide range of differential and variational problems.
Paper discovers differential equations from data using neural networks and Bayesian methods.
problem Discovering differential equations from datasets using machine learning.
method Integrates neural network-based surrogates with Sparse Bayesian Learning (SBL).
result Proposes a robust model discovery algorithm and a Physics Informed Normalizing Flow (PINF).
Unified bounds for neural networks incorporating physical laws.
problem Limitations in existing generalization analyses for PINNs and VPINNs.
method Unified framework using Taylor expansion and Koopman-based analysis.
result High-rank networks can generalize well even with differential operators.
Study shows AD for neural nets with machine-representable numbers can be incorrect.
problem Correctness of AD for neural nets with machine-representable numbers.
method Analyzed two sets of parameters: incorrect and non-differentiable. Proved bounds and conditions for AD correctness.
result AD can be incorrect for machine-representable numbers, but provides a Clarke subderivative on non-differentiable set.
Physics-informed neural networks approximate diffusion process pdfs efficiently.
problem Approximating the probability density function of diffusion processes.
method Physics-informed neural networks solving Fokker-Planck or integro-differential equations.
result Neural network solutions approximate target solutions for various types of differential equations.
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.
Continuous neural networks using differential equations.
problem Training and optimizing neural networks with continuous-depth architectures.
method Parameterize derivative of hidden state using neural networks and solve differential equations.
result End-to-end training of continuous-depth models.
Deep neural networks solve complex geometry PDEs.
problem Solving PDEs in complex geometries.
method Modified backpropagation for complex geometries, gradient descent or quasi-Newton optimization.
result Deep neural networks can approximate solutions to PDEs in complex geometries.
Graph neural networks are extended to continuous-depth models using differential equations.
problem Improving graph neural networks for static and dynamic graph data.
method Formalizing GNNs as GDEs, blending discrete structures with differential equations.
result GDEs offer computational advantages in static settings and improved performance in dynamic settings.
Efficient echo state network with explicit memory performs well on benchmark tasks.
problem Training differentiable neural computers is difficult and time-consuming.
method Echo state network with an explicit memory.
result Echo state network can recognize all regular languages, including those contractive networks cannot.
Neural differential equations combine deep learning and differential equations for modeling complex systems.
problem Modeling complex systems with high capacity and efficiency.
method Combining neural networks and differential equations, focusing on neural ordinary, controlled, and stochastic differential equations.
result NDEs offer high-capacity function approximation, strong priors, and handle irregular data efficiently.
New method improves training stochastic neural networks with tighter guarantees.
problem Training stochastic neural networks with provable guarantees.
method Developed partially-aggregated estimators and reformulated PAC-Bayesian bounds.
result Derives a differentiable objective leading to tighter generalisation guarantees.
Neural GDEs improve graph prediction by blending discrete structures and differential equations.
problem Dynamic graph prediction challenges in irregularly sampled data.
method Continuous-depth graph neural networks (GNNs) with Neural GDEs.
result Neural GDEs enhance performance across various applications.
We develop a scalable method for Bayesian neural networks with stochastic differential equations.
problem Uncertainty quantification in deep neural networks.
method Gradient-based stochastic variational inference in continuous-depth Bayesian neural networks.
result Gradient estimator with zero variance as the approximation improves.
A new method to simplify deep neural networks by removing unnecessary parts.
problem Overly complex deep neural networks require significant resource investment for size reduction.
method A fully differentiable sparsification method that optimizes a regularized objective function with stochastic gradient descent.
result The method can learn both the sparsified structure and weights of a network in an end-to-end manner.
Neural networks model financial data with Lévy processes.
problem Forecasting chaotic financial time series with big jumps.
method Lévy-induced stochastic differential equation network approximated by neural networks.
result The method improves prediction accuracy using non-Gaussian Lévy processes.
Neural networks can solve complex PDEs with minimal parameters.
problem Using neural networks to solve partial differential equations.
method Investigated two PDEs: Poisson and steady Navier--Stokes. Analyzed neural network architecture, initialization, loss function, and compared to classical methods.
result Small neural networks (<500 learnable parameters) can accurately solve complex PDEs.
Differentially private dropout technique preserves privacy in neural network training.
problem Preserving privacy in large datasets used for neural network training.
method Introduces a Bayesian dropout technique that adds intrinsic noise for regularization and differential privacy.
result Demonstrates that the iterative nature of neural network training can be handled with a relaxed differential privacy concept.
Proposes Neural SDE for better model robustness and generalization.
problem Missing regularization mechanisms in Neural ODE networks.
method Integrates various regularization mechanisms via stochastic noise injection.
result Improves robustness and generalization compared to Neural ODE.
Upper bounds on neural network complexity for PDE solutions.
problem Approximating solutions of parametric PDEs without knowing their exact form.
method Using low-dimensionality of solution manifolds and a small reduced basis.
result Neural networks can approximate PDE solutions with sizes dependent only on the reduced basis.
Neural sorting network improves sorting accuracy for complex inputs.
problem Sorting complex inputs like multi-digit images.
method Differentiable swap function and permutation-equivariant Transformer network.
result Our method outperforms baseline methods on various sorting benchmarks.
VSDN models sporadic time series with neural SDEs.
problem Modeling irregular and sparse time series data.
method Variational Bayesian method and neural SDEs.
result VSDNs outperform state-of-the-art models in prediction and interpolation.
Improves discrete latent representations using differentiable approximation bridges.
problem Improving discrete latent representations in neural networks.
method Training with a differentiable approximation bridge (DAB) neural network.
result Improves state-of-the-art performance in various domains.
ICON learns differential equation operators from prompts, reducing retraining and improving few-shot learning.
problem Training neural networks to solve differential equations without retraining for new problems.
method In-Context Operator Networks (ICON) that learns operators from prompted data and applies them to new problems.
result ICON can generalize to new operators beyond the training distribution and requires only a few demos.
A new differential entropy estimator for neural networks training.
problem Lack of effective differential entropy estimators for neural network training.
method KNIFE: a fully parameterized, differentiable kernel-based estimator of differential entropy.
result KNIFE effectively estimates differential entropy and improves neural network training.
This paper extends geometric study of neural networks to non-differentiable layers and random walks.
problem Understanding the geometric properties of neural networks, especially those with non-differentiable activation functions.
method Singular Riemannian geometry approach to convolutional, residual, and recursive neural networks.
result Illustrated geometric findings with numerical experiments on image classification and thermodynamic problems.
New framework establishes positivity of DNTK for PINNs.
problem Establishing positivity of NTK for PINNs with multiple differential operators.
method Proposed Differential Neural Tangent Kernel (DNTK) for PINNs.
result Positivity of infinite width DNTK for various activation functions and differential operators.
DEQGAN uses GANs to solve differential equations without supervision.
problem Solving differential equations with neural networks.
method Generative Adversarial Networks (GANs) to learn the loss function.
result DEQGAN achieves lower mean squared errors and competitive solution accuracy compared to traditional methods.
Neural networks trained with Q-learning converge to optimal control.
problem Optimal control of reinforcement learning with neural networks.
method Analysis of a limit differential equation derived from Q-learning.
result The limit differential equation has a unique stationary solution that is optimal.
New method solves PDEs for any initial condition without retraining.
problem Solving PDEs for different initial conditions requires retraining neural solvers.
method Formulate solution as conditional probability distribution.
result Approximates PDE solution for arbitrary initial conditions.
Normalization layers improve the accuracy of Differentially Private training of deep neural networks.
problem Reduced accuracy in deep neural networks with Differentially Private training.
method Proposed a novel method for integrating batch normalization with Differentially Private Stochastic Gradient Descent (DPSGD) without additional privacy loss.
result Training deeper networks with better utility-privacy trade-off is possible.
Hybrid model combines neural networks and fluid dynamics for efficient, generalized simulations.
problem Inefficient and poor generalization of deep learning approximations of fluid dynamics.
method Combines graph neural networks with a differentiable PDE solver inside a neural network.
result Hybrid model generalizes well to new scenarios and outperforms both neural network and traditional methods.
Proposes a method to train neural networks that solve differential equations faster.
problem Training neural networks that solve differential equations becomes computationally expensive.
method Introduces a differentiable surrogate for numerical solver time cost using higher-order derivatives.
result Trains models that are faster to solve while maintaining nearly the same accuracy.
DEUs learn nonlinear activation functions from data, reducing network size.
problem Fixed activation functions in neural networks limit performance.
method Differential equation units (DEUs) learn nonlinear activation functions from data.
result DEUs enable neurons to change their activation functions during training.
Method uses neural networks to fit nonlinear operators from data.
problem Finding nonlinear integro-differential operators from data.
method Parametrizes spatial operator with neural networks and Fourier transforms.
result Can recover spatial operators in fractional heat and Kuramoto-Sivashinsky equations.
Paper presents an ADMM-based approach to efficiently integrate quadratic programming layers into neural networks.
problem Integrating quadratic programs into neural networks for optimization.
method An ADMM-based network layer architecture for solving quadratic programs efficiently.
result The ADMM layer is approximately an order of magnitude faster than existing methods for medium scaled problems.
Neural networks solve SPDEs using Wiener chaos expansion.
problem Solving stochastic partial differential equations (SPDEs) numerically.
method Using neural networks in the truncated Wiener chaos expansion.
result Approximation rates for learning SPDE solutions with noise.
Neural networks learn general representations for solving differential equations.
problem Solving parametrized boundary value problems using neural networks.
method Singular Vector Canonical Correlation Analysis (SVCCA) for measuring generality.
result First hidden layer learns general representations, deeper layers become more specific.
FDNet learns PDEs from data with fast predictions.
problem Discovering complex systems behavior from data.
method Finite difference neural networks (FDNet) to learn PDEs from trajectory data.
result FDNet predicts future behavior with few trainable parameters.