Physics-informed dynamic mode decomposition (piDMD) (2112.04307v1)

Abstract: In this work, we demonstrate how physical principles -- such as symmetries, invariances, and conservation laws -- can be integrated into the dynamic mode decomposition (DMD). DMD is a widely-used data analysis technique that extracts low-rank modal structures and dynamics from high-dimensional measurements. However, DMD frequently produces models that are sensitive to noise, fail to generalize outside the training data, and violate basic physical laws. Our physics-informed DMD (piDMD) optimization, which may be formulated as a Procrustes problem, restricts the family of admissible models to a matrix manifold that respects the physical structure of the system. We focus on five fundamental physical principles -- conservation, self-adjointness, localization, causality, and shift-invariance -- and derive several closed-form solutions and efficient algorithms for the corresponding piDMD optimizations. With fewer degrees of freedom, piDMD models are less prone to overfitting, require less training data, and are often less computationally expensive to build than standard DMD models. We demonstrate piDMD on a range of challenging problems in the physical sciences, including energy-preserving fluid flow, travelling-wave systems, the Schr\"odinger equation, solute advection-diffusion, a system with causal dynamics, and three-dimensional transitional channel flow. In each case, piDMD significantly outperforms standard DMD in metrics such as spectral identification, state prediction, and estimation of optimal forcings and responses.

• The paper introduces piDMD, integrating fundamental physical principles as matrix manifold constraints into the Dynamic Mode Decomposition framework via a Procrustes problem formulation.
• Numerical studies show piDMD improves spectral identification, state prediction, and noise resilience across physical applications like fluid dynamics and quantum mechanics compared to standard DMD.
• piDMD provides a robust, noise-resistant tool for modeling complex systems and bridges machine learning with physics by embedding domain knowledge for better generalization.

An Examination of Physics-Informed Dynamic Mode Decomposition (piDMD)

This paper presents a structured approach to integrating physical principles into dynamic mode decomposition (DMD), termed physics-informed DMD (piDMD). DMD is a pivotal technique in data analysis, acclaimed for its ability to discern coherent dynamics and extract low-rank modal structures from high-dimensional datasets. However, standard DMD often falls short due to its susceptibility to noise, failure to generalize beyond training data, and violation of fundamental physical principles. The authors propose piDMD as a solution to these shortcomings by constraining the learned models to a matrix manifold that respects the physical laws governing the system at hand.

Key Concepts and Methodology

Central to this research is the incorporation of five fundamental physical principles into the DMD framework: conservation laws, self-adjointness, localization, causality, and shift-invariance. Each principle corresponds to specific structural constraints on the model matrix, embracing physically significant matrices such as unitary, symmetric, or tridiagonal forms. Such constraints are operationalized using a reformulation of the DMD regression task as a Procrustes problem, allowing the imposition of these physical structures via matrix manifold constraints.

The paper outlines the necessary steps for implementing piDMD, from model formulation and interpretation of known physical laws into matrix constraints, to the optimization problem and diagnostic evaluation of the computed models. Notably, the authors present several closed-form solutions to these optimization problems, employing techniques from linear algebra to efficiently enforce these constraints on large-scale data.

Numerical Results and Implications

The paper offers compelling numerical studies demonstrating piDMD's enhanced performance across several applications in the physical sciences, from fluid dynamics to quantum mechanics. In fluid dynamics, for instance, piDMD excels in spectral identification and state prediction for energy-preserving systems by embedding conservation laws into the model structure. Similar successes are highlighted in quantum systems, where piDMD skillfully captures the eigenstates of an unknown Hamiltonian, yielding real and positive energy levels consistent with quantum theory.

A notable feature of this work is the extensive sensitivity analysis the authors conduct to assess piDMD's resilience to noise, a common obstacle in data-driven modeling. By adopting noise-robust structures such as unitary or symmetric matrices, piDMD demonstrates reduced sensitivity compared to traditional DMD techniques.

Theoretical and Practical Impacts

The implications of piDMD are twofold—practical and theoretical. Practically, piDMD provides a robust, noise-resistant tool that better aligns with the true physics of the systems being modeled. This improvement can lead to more reliable model predictions and insights, crucial for systems where understanding dynamic behavior is pivotal, such as aerospace engineering or climate modeling.

Theoretically, piDMD bridges a key gap between machine learning and physics, showcasing how data-driven modeling can be significantly enhanced by embedding domain-specific knowledge. The framework offers a way to leverage partial knowledge about a system's physical laws, thus enabling modeling with fewer data requirements and improved generalization.

Future Directions

The work sets the stage for several future research directions. One potential avenue is exploring more complex matrix constraints or manifold structures, which could capture more intricate physical behaviors or symmetries. Additionally, extending piDMD to real-time, adaptive systems that update their constraints dynamically in response to new data could be transformative, particularly in applications requiring fast response to changing conditions.

Moreover, while the focus here remains on linear constraints, nonlinear extensions or hybrid models that combine piDMD with nonlinear machine learning models might offer even broader applicability. Lastly, integrating piDMD within larger-scale simulation frameworks or alongside other data-driven techniques could open new frontiers in multi-physics modeling and simulation.

In summary, this paper presents a methodologically sound approach to marrying physics with machine learning, promising significant advances in both understanding and predicting complex dynamical systems.