# Neural Laplace: Learning diverse classes of differential equations in the Laplace domain

arXiv: https://arxiv.org/abs/2206.04843 · HTML: https://mlsanity.com/paper/2206.04843/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 62 references that are in the map.
- First posted on arXiv 10 Jun 2022.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Neural Flows: Efficient Alternative to Neural ODEs](https://mlsanity.com/paper/2110.13040/) (2021)
- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [STEER: Simple Temporal Regularization For Neural ODEs](https://mlsanity.com/paper/2006.10711/) (2020)
- [ODE2VAE: Deep generative second order ODEs with Bayesian neural networks](https://mlsanity.com/paper/1905.10994/) (2019)
- [Neural Delay Differential Equations](https://mlsanity.com/paper/2102.10801/) (2021)
- [Neural Ordinary Differential Equations for Intervention Modeling](https://mlsanity.com/paper/2010.08304/) (2020)
- [A time-causal and time-recursive scale-covariant scale-space representation of temporal signals and past time](https://mlsanity.com/paper/2202.09209/) (2022)
- [Delay Differential Neural Networks](https://mlsanity.com/paper/2012.06800/) (2020)

## Built on by (most cited first, 24 of 24)

- [Discovering Preference Optimization Algorithms with and for Large Language Models](https://mlsanity.com/paper/2406.08414/) (2024)
- [Rough Transformers: Lightweight and Continuous Time Series Modelling through Signature Patching](https://mlsanity.com/paper/2405.20799/) (2024)
- [Automatically Learning Hybrid Digital Twins of Dynamical Systems](https://mlsanity.com/paper/2410.23691/) (2024)
- [ODE Discovery for Longitudinal Heterogeneous Treatment Effects Inference](https://mlsanity.com/paper/2403.10766/) (2024)
- [Training Stiff Neural Ordinary Differential Equations with Implicit Single-Step Methods](https://mlsanity.com/paper/2410.05592/) (2024)
- [Neural Laplace Control for Continuous-time Delayed Systems](https://mlsanity.com/paper/2302.12604/) (2023)
- [From Fourier to Neural ODEs: Flow Matching for Modeling Complex Systems](https://mlsanity.com/paper/2405.11542/) (2024)
- [Training Stiff Neural Ordinary Differential Equations with Explicit Exponential Integration Methods](https://mlsanity.com/paper/2412.01181/) (2024)
- [LES-SINDy: Laplace-Enhanced Sparse Identification of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2411.01719/) (2024)
- [Learning PDE Solution Operator for Continuous Modeling of Time-Series](https://mlsanity.com/paper/2302.00854/) (2023)
- [Time and State Dependent Neural Delay Differential Equations](https://mlsanity.com/paper/2306.14545/) (2023)
- [The Vanishing Gradient Problem for Stiff Neural Differential Equations](https://mlsanity.com/paper/2508.01519/) (2025)
- [Disentangling Slow and Fast Temporal Dynamics in Degradation Inference with Hierarchical Differential Models](https://mlsanity.com/paper/2509.00639/) (2025)
- [Graph Fourier Neural ODEs: Modeling Spatial-temporal Multi-scales in Molecular Dynamics](https://mlsanity.com/paper/2411.01600/) (2024)
- [No Equations Needed: Learning System Dynamics Without Relying on Closed-Form ODEs](https://mlsanity.com/paper/2501.18563/) (2025)
- [Symbolic Neural Ordinary Differential Equations](https://mlsanity.com/paper/2503.08059/) (2025)
- [Mining Causality from Continuous-time Dynamics Models: An Application to Tsunami Forecasting](https://mlsanity.com/paper/2210.04958/) (2022)
- [A projected nonlinear state-space model for forecasting time series signals](https://mlsanity.com/paper/2311.13247/) (2023)
- [Breaking Free: Decoupling Forced Systems with Laplace Neural Networks](https://mlsanity.com/paper/2503.13158/) (2025)
- [Neural Mean-Field Games: Extending Mean-Field Game Theory with Neural Stochastic Differential Equations](https://mlsanity.com/paper/2504.13228/) (2025)
- [Learning non-Markovian Dynamical Systems with Signature-based Encoders](https://mlsanity.com/paper/2509.12022/) (2025)
- [Knowledge-Informed Kernel State Reconstruction from Heterogeneous Partial Observations](https://mlsanity.com/paper/2601.22328/) (2026)
- [Latent Laplace Diffusion for Irregular Multivariate Time Series](https://mlsanity.com/paper/2605.19805/) (2026)
- [Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis](https://mlsanity.com/paper/2609.04329/) (2026)

## Abstract (arXiv)

Neural Ordinary Differential Equations model dynamical systems with ODEs learned by neural networks. However, ODEs are fundamentally inadequate to model systems with long-range dependencies or discontinuities, which are common in engineering and biological systems. Broader classes of differential equations (DE) have been proposed as remedies, including delay differential equations and integro-differential equations. Furthermore, Neural ODE suffers from numerical instability when modelling stiff ODEs and ODEs with piecewise forcing functions. In this work, we propose Neural Laplace, a unified framework for learning diverse classes of DEs including all the aforementioned ones. Instead of modelling the dynamics in the time domain, we model it in the Laplace domain, where the history-dependencies and discontinuities in time can be represented as summations of complex exponentials. To make learning more efficient, we use the geometrical stereographic map of a Riemann sphere to induce more smoothness in the Laplace domain. In the experiments, Neural Laplace shows superior performance in modelling and extrapolating the trajectories of diverse classes of DEs, including the ones with complex history dependency and abrupt changes.