## Code for paper "Neural Laplace: Learning diverse classes of differential equations in the Laplace domain"

Abstract: 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.

# PyTorch Implementation of Differentiable Laplace Reconstructions

This library provides Inverse Laplace Transform (ILT) algorithms implemented in PyTorch. Backpropagation through differential equation (DE) solutions in the Laplace domain is supported using the Riemann stereographic projection for better global representation of the complex Laplace domain. For usage for DE representations in the Laplace domain in deep learning applications, see reference [1].

## Installation

To install latest stable version:

pip install torchlaplace


To install the latest on GitHub:

pip install git+https://github.com/samholt/torchlaplace


## Tutorials

1. Tutorial: Laplace Reconstruct
2. Tutorial: Inverse Laplace Transform Algorithms

## Examples

Examples are placed in the examples directory.

We encourage those who are interested in using this library to take a look at examples/simple_demo.py for understanding how to use torchlaplace to fit a DE system.

## Basic usage

This library provides one main interface laplace_reconstruct which uses a selected inverse Laplace transform algorithm to reconstruct trajectories from a provided parameterized Laplace representation functional $\mathbf{F}(\mathbf{p},\mathbf{s})$ ,

$$\mathbf{x}(t) = \text{inverse laplace transform}(\mathbf{F}(\mathbf{p},\mathbf{s}), t)$$

Where $\mathbf{p}$ is a Tensor encoding the initial system state as a latent variable, and $t$ is the time points to reconstruct trajectories for.

This can be used by

from torchlaplace import laplace_reconstruct

laplace_reconstruct(laplace_rep_func, p, t)


where laplace_rep_func is any callable implementing the parameterized Laplace representation functional $\mathbf{F}(\mathbf{p},\mathbf{s})$ , p is a Tensor encoding the initial state of shape $(\text{MiniBatchSize},\text{K})$ . Where $\text{K}$ is a hyperparameter, and can be set by the user. Finally, t is a Tensor of shape $(\text{MiniBatchSize},\text{SeqLen})$ or $(\text{SeqLen})$ containing the time points to reconstruct the trajectories for.

Note that this is not numerically stable for all ILT methods, however should probably be fine with the default fourier (fourier series inverse) ILT algorithm.

The parameterized Laplace representation functional laplace_rep_func, $\mathbf{F}(\mathbf{p},\mathbf{s})$ also takes an input complex value $\mathbf{s}$ . This $\mathbf{s}$ is used internally when reconstructing a specified time point with the selected inverse Laplace transform algorithm ilt_algorithm.

The biggest gotcha is that laplace_rep_func must be a nn.Module when using the laplace_rep_func function. This is due to internally needing to collect the parameters of the parameterized Laplace representation.

To replicate the experiments in [1] see the in the experiments directory.

### Keyword arguments for laplace_rep_func

#### Keyword arguments:

• recon_dim (int): trajectory dimension for a given time point. Corresponds to dim $d_{\text{obs}}$ . If not explicitly specified, will use the same last dimension of p, i.e. $\text{K}$ .
• ilt_algorithm (str): inverse Laplace transform algorithm to use. Default: fourier. Available are {fourier, dehoog, cme, fixed_tablot, stehfest}. See api documentation on ILTs for further details.
• use_sphere_projection (bool): this uses the laplace_rep_func in the stereographic projection of the Riemann sphere. Default True.
• ilt_reconstruction_terms (int): number of ILT reconstruction terms, i.e. the number of complex $s$ points in laplace_rep_func to reconstruct a single time point.

#### List of ILT Algorithms:

ILT algorithms implemented:

• fourier Fourier Series Inverse [default].
• dehoog DeHoog (Accelerated version of Fourier) - Slower inference in comparison.
• cme Concentrated Matrix Exponentials.
• fixed_tablot Fixed Tablot.
• stehfest Gaver-Stehfest.

For most problems, good choices are the default fourier. However other ILT algorithms may be more appropriate when using higher ILT reconstruction terms, such as the cme algorithm. Some allow trade-offs between speed and accuracy, for example dehoog is very accurate if the representation is known or exact, however is slow and can be unstable to use when learning the correct representation.

## Detailed documentation

For detailed documentation see the official docs.

Take a look at our FAQ for frequently asked questions.

## References

For usage for DE representations in the Laplace domain and leveraging the stereographic projection and other applications see:

[1] Samuel Holt, Zhaozhi Qian, and Mihaela van der Schaar. "Neural laplace: Learning diverse classes of differential equations in the laplace domain." International Conference on Machine Learning. 2022. [arxiv]

If you found this library useful in your research, please consider citing.

@inproceedings{holt2022neural,
title={Neural Laplace: Learning diverse classes of differential equations in the Laplace domain},
author={Holt, Samuel I and Qian, Zhaozhi and van der Schaar, Mihaela},
booktitle={International Conference on Machine Learning},
pages={8811--8832},
year={2022},
organization={PMLR}
}