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On Neural Differential Equations
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The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.
Forward citations
Cited by 16 Pith papers
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Pathwise Learning of Stochastic Dynamical Systems with Partial Observations
A pathwise Zakai-equation control formulation is used to train conditional neural SDEs that amortize nonlinear filtering of partially observed stochastic dynamics.
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End-to-end differentiable retrieval of molecular spectra using hydrodynamics, chemistry, and radiative transfer
An end-to-end differentiable JAX pipeline couples 1D hydrodynamics, time-dependent chemistry, and radiative transfer, and recovers shock and rate parameters from synthetic HCO+ spectra.
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An Adjoint-Based Differentiable Physics Framework for Online Parameter Inversion in Closed-Brayton Gas-Cooled Reactor Digital Twins
Reverse-mode automatic differentiation through a closed-Brayton reactor DAE twin makes gradient-based parameter inversion match or beat Kalman filters on transient and partial-observation benchmarks, with 0.43% mean e...
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CausticFlow: An Efficient Machine Learning Framework Combining Neural Differential Equations and Normalizing Flows for Binary Microlensing Parameter Inference
CausticFlow combines neural CDEs and normalizing flows to propose binary microlensing posteriors in under a second, recovering ~80% of simulated events and 7 of 10 real events after local polishing.
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A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Series
The Weak Penalty Neural ODE uses a weak form loss to filter noise and learn stable chaotic dynamics from noisy observations.
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Generalization Bound for a General Class of Neural Ordinary Differential Equations
Claims a first generalization bound for nonlinear neural ODEs, but bounds the complexity of time trajectories rather than input-output maps, leaving the main theorem unproven.
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To Trade or Not to Trade: An Agentic Approach to Estimating Market Risk Improves Trading Decisions
LLM-discovered stochastic models of price paths provide risk metrics that improve trader-agent decisions, raising average Sharpe ratios from 0.88 to 1.40 in the paper's backtests.
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Time Resolution Independent Operator Learning
A DeepONet with a neural controlled differential equation branch and a trunk that takes space and time as inputs predicts transient mechanical fields from load histories at arbitrary spatiotemporal query points.
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Operator learning for models of tear film breakup
Operator learning can approximate the inverse mapping from fluorescence intensity to tear film thickness and osmolarity on synthetic data, but predictions diverge from ODE-based reference fits on experimental data.
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Understanding Malware Propagation Dynamics through Scientific Machine Learning
A hybrid physics-neural model (Universal Differential Equation) fits Code Red worm data with lower error than pure ODE or neural baselines, but the result is weakened because the model uses the observed data as an ext...
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Mass Transfer Through Vapor-Liquid Interfaces From Hydrodynamic Density Functional Theory
Hydrodynamic DFT reproduces NEMD results for mass transfer of an inserted second component across vapor-liquid interfaces for two LJTS mixtures at two temperatures.
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rodeo: Probabilistic Methods of Parameter Inference for Ordinary Differential Equations
rodeo is a JAX-based Python library that implements probabilistic ODE solvers and several Bayesian parameter inference methods with linear scaling in system size.
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Surrogate models for Rock-Fluid Interaction: A Grid-Size-Invariant Approach
Fully convolutional surrogate models trained on 64×64 patches predict 256×256 reactive-flow fields with competitive accuracy and lower GPU memory than full-domain or reduced-order models.
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Data-driven modeling of a settling sphere in a quiescent medium
Neural ODE and neural SDE models trained on experimental particle tracks reproduce long-time statistics of a chaotic settling sphere, with deterministic models generalizing better to new initial conditions.
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An Agentic AI Workflow to Simplify Parameter Estimation of Complex Differential Equation Systems
An agentic AI workflow converts XML problem specs and Python skeletons into JIT-compiled, differentiable ODE parameter estimation pipelines using PSO plus gradient refinement.
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Deep Learning in Classical and Quantum Physics
A graduate-level lecture-note review of deep learning methods and their applications in classical and quantum physics, with hands-on examples.
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