Enhancing Symbolic Regression with Nested Neural Networks: A Generalized SINDy Approach

Core Concepts

The Nested SINDy approach introduces additional layers before and after the core SINDy layer to increase the expressivity of the SINDy framework, enabling the identification of symbolic representations for a wider range of systems, including those with compositions and products of functions.

Abstract

The paper introduces the Nested SINDy approach, which aims to enhance the expressivity of the Sparse Identification of Nonlinear Dynamical Systems (SINDy) framework for symbolic regression.
The key highlights are:

The traditional SINDy method struggles with complex systems that cannot be easily described analytically. To address this, the Nested SINDy approach introduces additional layers before and after the core SINDy layer.

Two Nested SINDy architectures are proposed: the PR (Polynomial-Radial) model and the PRP (Polynomial-Radial-Polynomial) model. These models can capture more complex relationships in the data, such as compositions and products of functions.

The training process for the Nested SINDy models is more challenging due to the increased complexity of the optimization landscape. Strategies like adaptive Lasso regularization, pruning, and gradient noise injection are employed to stabilize the training.

Experiments on function discovery and ODE discovery tasks demonstrate the ability of the Nested SINDy approach to accurately identify symbolic representations for a wider range of systems compared to the traditional SINDy method.

The Nested SINDy framework offers new opportunities for understanding complex systems through data-driven methods, but also highlights the need for further research to develop more robust optimization techniques.

Stats

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Quotes

"Symbolic Regression (SR) is a widely studied field of research that aims to infer symbolic expressions from data."
"The rise of neural networks and auto-differentiation has spurred significant efforts to incorporate these techniques into SR, challenging the dominance of Eureqa-like approaches."
"SINDy-like approaches are the only type of unsupervised techniques capable of directly utilizing gradients from data to iteratively refine function expressions as they effectively render the discrete symbolic optimization problem continuous."

Key Insights Distilled From

Generalizing the SINDy approach with nested neural networks

by Cami... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15742.pdf

Deeper Inquiries

How can the Nested SINDy approach be extended to handle systems of ODEs or PDEs?

The Nested SINDy approach can be extended to handle systems of Ordinary Differential Equations (ODEs) or Partial Differential Equations (PDEs) by incorporating multiple dependent variables and their corresponding derivatives. In the context of ODEs, the system can be represented as a set of coupled equations involving multiple variables and their respective derivatives with respect to a single independent variable. The Nested SINDy framework can be adapted to identify the governing equations for each variable by introducing additional layers in the neural network architecture to capture the interactions and dependencies among the variables.
For systems of PDEs, the approach can be further expanded to include spatial derivatives and multiple dimensions. By incorporating spatial coordinates and their derivatives, the Nested SINDy model can learn the underlying dynamics of the system and identify the partial differential equations governing the evolution of the system in space and time. The neural network structure can be modified to accommodate the additional complexity of PDEs, allowing for the discovery of the underlying equations describing the system's behavior.

What are the theoretical guarantees on the convergence and optimality of the Nested SINDy optimization process?

The theoretical guarantees on the convergence and optimality of the Nested SINDy optimization process depend on the specific optimization algorithms and regularization techniques employed during training. While there may not be strict theoretical guarantees due to the complexity of the optimization landscape, certain strategies can enhance the convergence and optimality of the Nested SINDy framework.

Regularization Techniques: The use of regularization methods such as Lasso regularization can promote sparsity in the model, leading to simpler and more interpretable solutions. By controlling the complexity of the model, regularization can prevent overfitting and improve generalization performance.

Optimization Algorithms: The choice of optimization algorithms, such as Adam, stochastic gradient descent, or LBFGS, can impact the convergence of the optimization process. Adaptive learning rates, momentum, and other optimization parameters can influence the speed and stability of convergence.

Pruning and Noise Injection: Techniques like weight pruning and adding noise to the gradient can help the model escape local minima and explore the solution space more effectively. These strategies can prevent the model from getting stuck in suboptimal solutions and improve convergence towards better solutions.

While there may not be strict theoretical guarantees, the combination of these techniques can enhance the convergence and optimality of the Nested SINDy optimization process, leading to more robust and accurate results.

Can the Nested SINDy framework be combined with domain-specific knowledge to further improve its performance on complex real-world problems?

Yes, the Nested SINDy framework can be combined with domain-specific knowledge to enhance its performance on complex real-world problems. By incorporating domain expertise, constraints, or insights into the model architecture and training process, the framework can leverage prior knowledge to guide the discovery of symbolic representations of dynamical systems.

Feature Engineering: Domain-specific knowledge can be used to design the dictionary of functions or basis set tailored to the specific problem domain. By including relevant functions or constraints based on domain expertise, the model can focus on capturing the essential dynamics of the system.

Constraint Incorporation: Domain-specific constraints or physical laws can be integrated into the optimization process to guide the model towards solutions that adhere to known principles. By imposing constraints on the model parameters or structure, the Nested SINDy framework can produce more meaningful and physically interpretable results.

Interpretability and Explainability: By incorporating domain-specific knowledge, the Nested SINDy model can generate more interpretable and explainable results. Domain experts can provide insights into the significance of the discovered equations and validate the model's predictions based on their expertise.

Overall, the integration of domain-specific knowledge can significantly improve the performance and applicability of the Nested SINDy framework in solving complex real-world problems by leveraging expert insights and constraints specific to the problem domain.