Inverse Design with Dynamic Mode Decomposition

TL;DR

This work tackles the computational bottleneck of inverse design for dynamic systems by introducing ID-DMD, a data-driven yet physics-grounded approach that builds a low-rank, linear-in-parameters model across design parameters. By leveraging least-squares regression and Koopman-inspired observables, ID-DMD achieves fast, interpretable design, robust to noise, with uncertainty quantification and scalable computation via randomized SVD. The method demonstrates superior speed and accuracy relative to leading operator-learning techniques, exemplified through airfoil angle optimization and a broad suite of dynamic systems, and shows strong extrapolation capabilities. The practical impact is a paradigm that enables laptop-scale, hardware-friendly inverse design while preserving physical insight and reliable predictions across parameter and time horizons.

Abstract

We introduce a computationally efficient method for the automation of inverse design in science and engineering. Based on simple least-square regression, the underlying dynamic mode decomposition algorithm can be used to construct a low-rank subspace spanning multiple experiments in parameter space. The proposed inverse design dynamic mode composition (ID-DMD) algorithm leverages the computed low-dimensional subspace to enable fast digital design and optimization on laptop-level computing, including the potential to prescribe the dynamics themselves. Moreover, the method is robust to noise, physically interpretable, and can provide uncertainty quantification metrics. The architecture can also efficiently scale to large-scale design problems using randomized algorithms in the ID-DMD. The simplicity of the method and its implementation are highly attractive in practice, and the ID-DMD has been demonstrated to be an order of magnitude more accurate than competing methods while simultaneously being 3-5 orders faster on challenging engineering design problems ranging from structural vibrations to fluid dynamics. Due to its speed, robustness, interpretability, and ease-of-use, ID-DMD in comparison with other leading machine learning methods represents a significant advancement in data-driven methods for inverse design and optimization, promising a paradigm shift in how to approach inverse design in practice.

Figures (12)

• Figure 1: The inverse design of airfoil pitched angle using ID-DMD. (a) Collect vorticity simulation data as snapshots across varying pitched angles $\theta$, with additional noise levels up to 15%. (b) Identify ID-DMD to capture the underlying dynamics of the system. (c) Perform model validation by evaluating vorticity predictions at different pitched angles. (d) Utilize the validated ID-DMD for airfoil pitched angle optimization. The design target is to achieve a preferred wavelength $\lambda$, while minimizing the airflow power ${P}_\text{air}$ across the fluid filed within the blue dotted box. (e) The design is robust to noise and stable with narrow uncertainty boundaries.
• Figure 2: Interpolation and extrapolation of the 1-D Burgers’ equation using the ID-DMD, PI-DON, PINNs, NIF, and FNO. (a) Interpolation prediction results for $v=0.02$ and $t\in [0,1]\text{s}$ using different advanced data-driven methods. (b) Relative errors for the interpolation results. (c) Extrapolation prediction results for $v=0.01$ and $t\in [0,3]\text{s}$ using different advanced data-driven methods. (d) Relative errors for the extrapolation results.
• Figure 3: Evaluate the first three order modes from the polynomial-projected Koopman operator. (a) The dominant modes and their corresponding characteristic frequencies can be identified from the imaginary part of the eigenvalues in the ID-DMD framework. These dominant modes remain stable across varying design parameters, representing the true dynamics of the system. In contrast, other modes that do not exhibit this stability are classified as spurious modes. (b) The modes of the system with (i) The first mode arises from the linear states, corresponding to the natural frequency of $\omega_\text{e}=10\ \text{rad/s}$. (ii) The second mode results from the squared projection of the states, occurring at the second-order modulation frequency $\omega_\text{e} =20\ \text{rad/s}$. (iii) The third mode, at $\omega_\text{e}=30\ \text{rad/s}$, is contributed by both the linear states and cubic projection of the states. (c) System response spectrum of $y(k)$ under $c_3=15$ and an input excitation $u(t)=\cos(10t)$. (d) Design the energy dissipation of ${\eta}_{\text{E}}>30\%$ with ${{c}_{3}}>12.6$
• Figure 4: Configuration for the vorticity simulation of a pitched airfoil.
• Figure 5: Prediction of the vorticity distribution around the airfoil.