ADAM-SINDy: Enhancing Sparse Identification of Nonlinear Dynamics with Adaptive Optimization for Parameter Estimation

Concetti Chiave

ADAM-SINDy improves upon the traditional SINDy algorithm by incorporating the ADAM optimization algorithm, enabling simultaneous identification of governing equations and estimation of nonlinear parameters in dynamical systems.

Sintesi

• Bibliographic Information: Viknesh, S., Tatari, Y., & Arzani, A. (2024). ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identification. arXiv preprint arXiv:2410.16528.
• Research Objective: This paper introduces ADAM-SINDy, a novel method for identifying parameterized nonlinear dynamical systems by integrating the ADAM optimization algorithm within the Sparse Identification of Nonlinear Dynamics (SINDy) framework.
• Methodology: ADAM-SINDy leverages the ADAM optimizer to simultaneously optimize nonlinear parameters and coefficients associated with candidate functions in a predefined library. The method employs a global optimization strategy, allowing for adaptive adjustment of parameters based on system-specific data. The performance of ADAM-SINDy is evaluated using a range of benchmark dynamical systems, including oscillators, chaotic fluid flows, reaction kinetics, pharmacokinetics, and a wildfire transport model.
• Key Findings: ADAM-SINDy demonstrates superior performance compared to the classical SINDy approach, particularly in systems characterized by nonlinear parameters. It accurately identifies governing equations and estimates nonlinear parameters without prior knowledge, addressing a key limitation of traditional SINDy. The method's adaptability and efficiency are highlighted through its ability to handle complex, high-dimensional systems.
• Main Conclusions: ADAM-SINDy presents a significant advancement in nonlinear system identification by enabling efficient and precise parameter estimation alongside the identification of governing equations. This capability enhances the applicability of the SINDy framework to a broader range of complex dynamical systems.
• Significance: This research contributes to the field of data-driven system identification by providing a robust and interpretable method for uncovering the underlying dynamics of complex systems. The integration of ADAM optimization within the SINDy framework offers a promising avenue for tackling challenges in various scientific and engineering domains.
• Limitations and Future Research: While ADAM-SINDy demonstrates promising results, further exploration of its capabilities in handling noisy data and high-dimensional systems is warranted. Future research could investigate the integration of physics-informed constraints within the ADAM-SINDy framework to enhance its accuracy and robustness in specific application domains.

Statistiche

The cosine frequency (d) in the harmonic oscillator example is set to 0.75.
The Van der Pol oscillator example uses µ = 0.01 and b = 2.15.
The ABC flow example incorporates six distinct frequencies: π/2, π/2.8, π/3, π/4, π/4.5, and π/5, with amplitudes A = 2, B = 3, and C = 1.
The chemical reaction kinetics example uses non-integer values for the parameters g = 1.015 and h = 1.025, with k = 0.07 and µ = 0.1.

Citazioni

Approfondimenti chiave tratti da

ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identification

by Siva Viknesh... alle arxiv.org 10-23-2024

https://arxiv.org/pdf/2410.16528.pdf

Domande più approfondite

How does the performance of ADAM-SINDy compare to other machine learning techniques for nonlinear system identification, such as symbolic regression or deep learning-based approaches, in terms of accuracy, efficiency, and interpretability?

ADAM-SINDy, symbolic regression, and deep learning each offer distinct advantages and disadvantages for nonlinear system identification. Here's a comparative analysis:
Accuracy:

ADAM-SINDy: Achieves high accuracy when the underlying dynamical system can be represented by a sparse combination of functions within its predefined library. However, its accuracy may be limited if the true governing equations involve functional forms not included in the library.
Symbolic Regression: Excels in discovering complex nonlinear relationships, potentially achieving high accuracy even for systems with unknown governing equations. Its flexibility comes at the cost of increased computational expense and the risk of overfitting.
Deep Learning: Can model highly complex systems with high accuracy, but often lacks interpretability. Deep learning models are often treated as black boxes, making it challenging to extract insights into the underlying physical processes.
Efficiency:

ADAM-SINDy: Computationally efficient, especially when leveraging stochastic gradient descent optimization. Its efficiency stems from the sparsity constraint and the use of a predefined library of candidate functions.
Symbolic Regression: Computationally intensive, particularly for high-dimensional systems or large datasets. The search for optimal symbolic expressions often requires significant computational resources.
Deep Learning: Can be computationally demanding, requiring substantial data and computational power for training. However, once trained, deep learning models can be highly efficient for inference.
Interpretability:

ADAM-SINDy: Yields highly interpretable models, as the identified governing equations are expressed as a sparse combination of functions from a predefined library. This interpretability makes it easier to understand the underlying physical phenomena.
Symbolic Regression: Produces interpretable models in the form of analytical expressions. However, the complexity of the identified equations can sometimes hinder interpretability, especially for highly nonlinear systems.
Deep Learning: Generally lacks interpretability, as the learned representations are often distributed and difficult to interpret directly. Techniques for interpreting deep learning models are an active area of research.
In summary:

ADAM-SINDy provides a balance of accuracy, efficiency, and interpretability, making it well-suited for systems where a sparse representation of the governing equations is expected.
Symbolic regression prioritizes accuracy and the ability to discover novel dynamical relationships, but at the cost of computational efficiency.
Deep learning offers high accuracy and can model highly complex systems, but often lacks interpretability.
The choice of technique depends on the specific application requirements, with a trade-off between accuracy, efficiency, and interpretability.

Could the reliance on a predefined library of candidate functions in ADAM-SINDy limit its ability to discover novel or unexpected dynamical relationships in real-world systems with unknown or highly complex governing equations?

Yes, the reliance on a predefined library of candidate functions in ADAM-SINDy can indeed limit its ability to discover novel or unexpected dynamical relationships in real-world systems with unknown or highly complex governing equations.
Here's why:

Limited Scope: The predefined library inherently restricts the search space for potential governing equations. If the true underlying relationship involves functional forms not present in the library, ADAM-SINDy will not be able to identify it.
Bias Towards Known Physics: The choice of library often reflects existing knowledge and assumptions about the system's behavior. This can introduce bias and prevent the discovery of truly novel dynamics that deviate from established understanding.
Difficulty in Representing Complex Interactions: Real-world systems often exhibit complex interactions that may not be easily captured by simple combinations of elementary functions. ADAM-SINDy's reliance on a limited set of functions can hinder its ability to model such intricate relationships.
Strategies to Mitigate Limitations:

Comprehensive Library Design: Carefully designing a comprehensive library that includes a wide range of functional forms can help mitigate this limitation. However, this can increase computational cost and may still not encompass all possible functional relationships.
Iterative Library Expansion: Starting with a smaller library and iteratively expanding it based on the model's performance can be a more adaptive approach. This allows for the incorporation of new functional forms as needed.
Hybrid Approaches: Combining ADAM-SINDy with other techniques, such as symbolic regression, can leverage the strengths of both approaches. For instance, symbolic regression could be used to suggest novel candidate functions to be added to the ADAM-SINDy library.
In essence: While ADAM-SINDy's reliance on a predefined library offers efficiency and interpretability, it can limit its ability to discover entirely novel dynamics. Addressing this limitation requires careful consideration of library design, potentially incorporating iterative or hybrid approaches to expand the search space and enhance its capacity to uncover hidden relationships in complex systems.

The paper highlights the potential of incorporating physics-informed constraints into ADAM-SINDy. How can this integration be effectively implemented, and what are the potential benefits and challenges of developing a fully physics-informed SINDy framework for practical applications?

Incorporating physics-informed constraints into ADAM-SINDy holds significant promise for enhancing its accuracy and physical consistency. Here's how it can be implemented and the potential benefits and challenges:
Implementation Strategies:

Regularization with Physics-Based Operators:

Incorporate differential operators (e.g., divergence, Laplacian) representing physical laws (e.g., conservation of mass, energy) into the sparsity-promoting term (Γ matrix in ADAM-SINDy).
Penalize candidate models that violate these physical constraints during optimization, guiding the search towards physically plausible solutions.

Physics-Informed Loss Function:

Augment the standard data-driven loss function with terms that penalize deviations from known physical equations or constraints.
For example, add a term that measures the residual of the governing PDE when evaluated with the candidate model.

Constrained Optimization:

Formulate the ADAM-SINDy optimization problem with explicit constraints that enforce physical laws.
Utilize constrained optimization algorithms to find solutions that satisfy both the data fidelity and the imposed physical constraints.

Potential Benefits:

Improved Accuracy and Physical Consistency: Physics-informed constraints guide the identification process towards models that adhere to known physical principles, leading to more accurate and physically meaningful representations of the system.
Reduced Data Requirements: Incorporating prior physical knowledge can compensate for limited or noisy data, as the constraints provide additional information to guide the model discovery process.
Enhanced Extrapolation Capabilities: Physics-informed models are more likely to generalize well beyond the training data range, as they capture the underlying physical principles governing the system's behavior.
Challenges:

Selection of Appropriate Constraints: Choosing relevant and informative physical constraints is crucial. Overly restrictive constraints can hinder the discovery of novel dynamics, while loosely defined constraints may not provide sufficient guidance.
Computational Complexity: Incorporating physics-informed constraints can increase the complexity of the optimization problem, potentially leading to longer training times.
Balancing Data Fidelity and Physical Consistency: Finding the right balance between fitting the observed data and satisfying the imposed physical constraints can be challenging. Trade-offs may be necessary depending on the specific application and data quality.
In conclusion: Developing a fully physics-informed SINDy framework requires addressing challenges related to constraint selection, computational complexity, and balancing data fidelity with physical consistency. However, the potential benefits, including improved accuracy, reduced data dependency, and enhanced extrapolation capabilities, make it a promising avenue for advancing system identification in various scientific and engineering domains.