On Neural Differential Equations

Kidger, Patrick

doi:10.48550/arxiv.2202.02435

Cited by 16 publications

(20 citation statements)

References 73 publications

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“…Neural Differential equations [38] provided an intimate survey on neural differential equations. The topic of neural ODEs becomes an emerging field since E [27] and Chen et al's work [19], with many follow-up works.…”

Section: Related Workmentioning

confidence: 99%

ACMP: Allen-Cahn Message Passing for Graph Neural Networks with Particle Phase Transition

Wang¹,

Yi²,

Liu³

et al. 2022

Preprint

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Neural message passing is a basic feature extraction unit for graph-structured data that takes account of the impact of neighboring node features in network propagation from one layer to the next. We model such process by an interacting particle system with attractive and repulsive forces and the Allen-Cahn force arising in the modeling of phase transition. The system is a reaction-diffusion process which can separate particles to different clusters. This induces an Allen-Cahn message passing (ACMP) for graph neural networks where the numerical iteration for the solution constitutes the message passing propagation. The mechanism behind ACMP is phase transition of particles which enables the formation of multiclusters and thus GNNs prediction for node classification. ACMP can propel the network depth to hundreds of layers with theoretically proven strictly positive lower bound of the Dirichlet energy. It thus provides a deep model of GNNs which circumvents the common GNN problem of oversmoothing. Experiments for various real node classification datasets, with possible high homophily difficulty, show the GNNs with ACMP can achieve state of the art performance with no decay of Dirichlet energy.

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Section: Related Workmentioning

confidence: 99%

ACMP: Allen-Cahn Message Passing for Graph Neural Networks with Particle Phase Transition

Wang¹,

Yi²,

Liu³

et al. 2022

Preprint

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“…Neural CDE approximates the underlying process of the time series as a differential equation (Kidger et al 2020;Kidger 2022)…”

Section: Methodsmentioning

confidence: 99%

MAGIC: Microlensing Analysis Guided by Intelligent Computation

Zhao

Zhu

2022

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The modeling of binary microlensing light curves via the standard sampling-based method can be challenging, because of the time-consuming light-curve computation and the pathological likelihood landscape in the high-dimensional parameter space. In this work, we present MAGIC, which is a machine-learning framework to efficiently and accurately infer the microlensing parameters of binary events with realistic data quality. In MAGIC, binary microlensing parameters are divided into two groups and inferred separately with different neural networks. The key feature of MAGIC is the introduction of a neural controlled differential equation, which provides the capability to handle light curves with irregular sampling and large data gaps. Based on simulated light curves, we show that MAGIC can achieve fractional uncertainties of a few percent on the binary mass ratio and separation. We also test MAGIC on a real microlensing event. MAGIC is able to locate degenerate solutions even when large data gaps are introduced. As irregular samplings are common in astronomical surveys, our method also has implications for other studies that involve time series.

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“…We build the estimator as the composition of two Neural Differential Equations (NDEs) [33]. The first NDE encodes the information carried by the filtration (F Y r ) 0≤r≤s , whilst the second NDE is carefully designed so that its vector field parametrizes the rate of change of the mean prospective transition, see Lemma 4.2 below.…”

Section: A Well-posedness Of the Estimatormentioning

confidence: 99%

Learning the conditional law: signatures and conditional GANs in filtering and prediction of diffusion processes

Germ¹,

Sabate-Vidales²

2022

Preprint

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We consider the filtering and prediction problem for a diffusion process. The signal and observation are modeled by stochastic differential equations (SDEs) driven by Wiener processes. In classical estimation theory, measurevalued stochastic partial differential equations (SPDEs) are derived for the filtering and prediction measures. These equations can be hard to solve numerically. We provide an approximation algorithm using conditional generative adversarial networks (GANs) and signatures, an object from rough path theory. The signature of a sufficiently smooth path determines the path completely. In some cases, GANs based on signatures have been shown to efficiently approximate the law of a stochastic process. In this paper we extend this method to approximate the prediction measure conditional to noisy observation. We use controlled differential equations (CDEs) as universal approximators to propose an estimator for the conditional and prediction law. We show well-posedness in providing a rigorous mathematical framework. Numerical results show the efficiency of our algorithm.

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On Neural Differential Equations

Cited by 16 publications

References 73 publications

ACMP: Allen-Cahn Message Passing for Graph Neural Networks with Particle Phase Transition

ACMP: Allen-Cahn Message Passing for Graph Neural Networks with Particle Phase Transition

MAGIC: Microlensing Analysis Guided by Intelligent Computation

Learning the conditional law: signatures and conditional GANs in filtering and prediction of diffusion processes

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