---
title: Laplace-Enhanced SINDy (LES-SINDy)
url: https://www.emergentmind.com/topics/laplace-enhanced-sindy-les-sindy
type: topic
---

# Laplace-Enhanced SINDy (LES-SINDy)

Laplace-Enhanced Sparse Identification of Nonlinear Dynamical Systems (LES-SINDy) is a data-driven framework for governing equation discovery that circumvents limitations of classical SINDy when faced with high-order derivatives, discontinuities, or unbounded growth functions, particularly under noisy conditions. LES-SINDy operates by transferring time-domain data to the Laplace domain, enabling analytic treatment of derivatives and discontinuous terms, and mitigating numerical difficulties in the identification of ordinary and partial differential equations (ODEs and PDEs) [2411.01719].

## 1. Theoretical Foundations

LES-SINDy extends the Sparse Identification of Nonlinear Dynamical Systems (SINDy) paradigm by integrating Laplace transform theory. For a vector-valued measurement time series $u(t)$ sampled at $t_1,\dots,t_m$, the method first constructs a SINDy library $X(t_j)$ that includes constants, $t_j$, $u(t_j)$, and derivatives up to order $k$, along with their tensor product monomials up to degree $n$. 

Each column $X_i(t)$ of the resulting $m\times d$ matrix undergoes a Laplace transform:
$$
\mathcal{L}\{X_i(t)\}(s) = \int_0^\infty e^{-st} X_i(t) \, dt,
$$
which, on finite data, is approximated by
$$
X_i(s) \approx \sum_{j=1}^m e^{-s(t_j-t_1)} X_i(t_j)\Delta t_j,
$$
with $\Delta t_j = t_{j+1} - t_j$ (except for $\Delta t_m = t_m - t_{m-1}$).

The Laplace transform treats derivatives analytically using integration by parts:
$$
\mathcal{L}\left\{\frac{d^k u}{dt^k}\right\}(s) = s^k \mathcal{L}\{u\}(s) - \sum_{n=0}^{k-1} s^{k-1-n} u^{(n)}(0).
$$
This enables LES-SINDy to avoid numerical differentiation entirely; all high-order derivatives are represented exactly in the transform domain if initial conditions are available.

## 2. Algorithmic Workflow

The LES-SINDy procedure is composed of three main steps:

1. **Laplace-Enhanced Library Construction:**
   - Assemble the candidate function matrix $X(t_j)$ from the time-series data.
   - Select $L$ complex frequencies $s_1,\dots,s_L$ with $\mathrm{Re}(s_\ell)>0$.
   - For each $\ell$, compute the Laplace domain vectors $X(s_\ell)$ as above; stack them to form $\Theta \in \mathbb{R}^{L \times d}$ with columns $X(s_\ell)$.

2. **Sparse Regression in the Laplace Domain:**
   - Solve the implicit equation $\Theta\xi = 0$ for $\xi\in\mathbb{R}^d$, seeking sparsity.
   - To avoid the trivial solution $\xi=0$, cycle through each $i$ and set $\xi_i=1$; then solve
     $$
     \min_{\zeta}\|\Theta_{-i}\zeta - \Theta_i\|_2^2 + \lambda R(\zeta)
     $$
     where $R(\zeta)$ is a sparsity-promoting regularizer, such as $\ell_1$ or thresholded least squares.
   - This process yields $d$ candidate models $\xi^{(i)}$.

3. **Model Evaluation and Selection:**
   - Each candidate $\xi^{(i)}$ is converted to an explicit ODE/PDE, integrated numerically (e.g., via Runge–Kutta), and produces a predicted trajectory $\hat{U}(t)$.
   - Two quality metrics are computed:
     - **log RMSE**:
       $$
       \varepsilon_1(\xi) = \log\left[\frac{1}{m}\sum_{j=1}^m \|u(t_j) - \hat{U}(t_j;\xi)\|_2^2\right]^{1/2}
       $$
       Used primarily for hyperparameter tuning.
     - **Corrected AIC (AICc):**
       $$
       \varepsilon_2(\xi) = 2p + m\ln(2\pi\sigma^2) + \frac{1}{\sigma^2}\sum_{j=1}^m \|u(t_j) - \hat{U}(t_j)\|^2 + \frac{2(p+1)(p+2)}{m-p-2}
       $$
       Where $p$ is the number of nonzero terms in $\xi$, and $\sigma^2$ the estimated residual variance. AICc is used for model selection.

The model minimizing AICc is selected as the optimal, parsimonious equation.

## 3. Analytic Handling of High-Order Derivatives and Discontinuities

A core innovation of LES-SINDy is its analytic approach for traditionally challenging features:

- **High-order derivatives:** By manipulating the Laplace transform, derivatives become algebraic (e.g., $s^k$ factors), avoiding finite-difference schemes that amplify noise.
- **Discontinuous and impulsive terms:** Step functions $H(t-t_0)$ and impulses $\delta(t-t_0)$ possess known Laplace transforms ($e^{-s t_0}/s$ and $e^{-s t_0}$, respectively), allowing explicit inclusion and identification.
- **Unbounded or slowly-growing functions:** The $e^{-s t}$ weighting in the Laplace transform de-emphasizes late times where numerical errors and exponential growth dominate, provided $\mathrm{Re}(s)$ is sufficiently large.

This analytic treatment broadens the class of systems addressable by SINDy methodologies.

## 4. Regression and Model Selection in the Laplace Domain

The transformation to the Laplace domain offers computational and statistical advantages:

- **Resolution-independent regression:** The feature matrix $\Theta$ is $L \times d$ and independent of the number of time samples $m$, decoupling regression cost from data resolution.
- Sparse regression is enforced in the Laplace domain, seeking a minimal support solution for $\xi$ that captures the underlying dynamical structure.
- After candidate equations are inferred, simulation and out-of-sample evaluation ensure that discovered models are both accurate and parsimonious, as quantified by the AICc metric.

## 5. Empirical Performance Across Dynamical Systems

LES-SINDy has been validated on a range of ODE and PDE identification tasks:

- **High-order ODEs:** Fourth-order models (e.g., $u_{tttt} + \alpha u_{tt} + \beta u = 0$) are recovered with exact coefficients even under substantial noise. LES-SINDy achieves substantial improvements in AICc (e.g., from $-587.8$ to $-2559.3$), while SINDy fails to resolve high-order derivatives or produces spurious terms.
- **Discontinuous forcing:** For ODEs with Heaviside or Dirac impulse terms, LES-SINDy accurately recovers both continuous and discontinuous dynamics. Classical SINDy fails or misestimates these terms, especially when noisy.
- **Trigonometric and hyperbolic forcing:** While Fourier-domain techniques are limited by unbounded inputs like $\sinh$ or $\cosh$, LES-SINDy successfully recovers their coefficients and structure, tolerating noise levels up to $\sim 10\%$.
- **Nonlinear ODE systems:** Canonical models such as Lorenz and Lotka–Volterra are reconstructed with parameter errors $<0.5\%$ and $<1\%$, respectively, at strongly negative AICc.
- **PDEs:** For convection–diffusion, Burgers, and Kuramoto–Sivashinsky equations, LES-SINDy recovers governing equations with $<3\%$ parameter error at noise levels up to $30\%$. It outperforms both classical SINDy and Weak-SINDy, tolerating up to twice the noise before failure in the Kuramoto–Sivashinsky test.

Table 1 below summarizes specific empirical findings.

| System Type       | Standard SINDy Limitation       | LES-SINDy Performance              |
|-------------------|--------------------------------|------------------------------------|
| High-order ODE    | Fails on $u_{tttt}$ with noise | Exact coefficient recovery, strong AICc gains |
| Discontinuities   | Misses $\delta,H$ terms        | Accurate recovery, $<1\%$ error under 10% noise |
| Unbounded forcing | Fourier methods break down     | Accurate recovery for $\sinh,\cosh$ |
| PDEs (KS, Burgers)| Low noise tolerance            | Twice noise tolerance, low parameter error |

## 6. Broader Implications and Outlook

LES-SINDy substantially extends the reach of sparse regression-based equation discovery in applied mathematics, physics, and engineering domains where discontinuities, high-order derivatives, and unbounded inputs are prevalent. Its computational efficiency due to resolution-independent regression and its robustness under noise suggest promise for both theoretical analysis and practical applications, including real-world scenarios with measurement challenges and incomplete initial conditions (*A plausible implication is that LES-SINDy may catalyze further development of Laplace-domain approaches to model inference*).

For implementation details, theoretical justification, and further benchmarks, see "LES-SINDy: Laplace-Enhanced Sparse Identification of Nonlinear Dynamical Systems" [2411.01719].

Source: https://www.emergentmind.com/topics/laplace-enhanced-sindy-les-sindy