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Sparse identification of nonlinear dynamics with low-dimensionalized flow representations

Kai Fukami, Takaaki Murata, Kai Zhang, Koji Fukagata

arXiv:2010.12177v2physics.flu-dyn

TL;DR

High-dimensional flow data pose a challenge for SINDy because its library is suited to combinations of low-dimensional variables. The paper combines a CNN autoencoder with SINDy to model latent flow dynamics and reconstruct flow fields, reporting successful wake modeling and equation recovery for a turbulent shear-flow model. The authors caution that CNN autoencoders may be unsuitable when many spatial modes are needed.

  • Problem

    SINDy's library is suited to low-dimensional variable combinations, motivating a way to model high-dimensional fluid flows through a lower-dimensional representation.

    Source: §1 · p. 2

  • Method

    A CNN autoencoder maps flow data into a smaller latent space, where SINDy identifies governing dynamics that the decoder maps back to flow fields.

    Source: §1 · p. 2, §2.2 · p. 4

  • Results

    With appropriate parameter choices, SINDy follows latent dynamics for the cylinder wake, its transient, and the two-parallel-cylinder wake, with required terms varying by complexity.

    Source: §4 · pp. 21–22

  • Takeaways & Limitations

    For the nine-equation turbulent shear-flow model, SINDy recovered the governing equation with an appropriate sparsity constant, supporting the possibility of interpretable turbulent-flow models if a suitable low-dimensional mapping is designed.

    Source: §4 · p. 22

    The current CNN autoencoder may struggle to compress flows requiring many spatial modes, including turbulence, so the low-dimensionalization tool should be chosen for the flow being modeled.

    Source: §4 · p. 22

Abstract

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We perform a sparse identification of nonlinear dynamics (SINDy) for low-dimensionalized complex flow phenomena. We first apply the SINDy with two regression methods, the thresholded least square algorithm (TLSA) and the adaptive Lasso (Alasso) which show reasonable ability with a wide range of sparsity constant in our preliminary tests, to a two-dimensional single cylinder wake at $Re_D=100$, its transient process, and a wake of two-parallel cylinders, as examples of high-dimensional fluid data. To handle these high dimensional data with SINDy whose library matrix is suitable for low-dimensional variable combinations, a convolutional neural network-based autoencoder (CNN-AE) is utilized. The CNN-AE is employed to map a high-dimensional dynamics into a low-dimensional latent space. The SINDy then seeks a governing equation of the mapped low-dimensional latent vector. Temporal evolution of high-dimensional dynamics can be provided by combining the predicted latent vector by SINDy with the CNN decoder which can remap the low-dimensional latent vector to the original dimension. The SINDy can provide a stable solution as the governing equation of the latent dynamics and the CNN-SINDy based modeling can reproduce high-dimensional flow fields successfully, although more terms are required to represent the transient flow and the two-parallel cylinder wake than the periodic shedding. A nine-equation turbulent shear flow model is finally considered to examine the applicability of SINDy to turbulence, although without using CNN-AE. The present results suggest that the proposed scheme with an appropriate parameter choice enables us to analyze high-dimensional nonlinear dynamics with interpretable low-dimensional manifolds.

1. Introduction

SINDy identifies interpretable governing equations from time-series data, and prior work has extended its use across fluid-dynamics problems. This study pairs SINDy with a CNN autoencoder to model several high-dimensional wakes through low-dimensional latent dynamics.

1. Introduction

  • SINDy identifies governing equations and dominant terms from time-discretized data, building on applications to experimental flows, wake dynamics, and turbulence modeling.Prior fluid-dynamics work also combines SINDy with control inputs, POD, or parametric PDE models.Source: §1 · pp. 1–2
  • The study applies TLSA and adaptive Lasso SINDy to a Re_D = 100 cylinder wake, its transient process, and a two-parallel-cylinder wake using a CNN autoencoder.The autoencoder maps high-dimensional flow data to a latent space; SINDy models its dynamics, and the decoder reconstructs the flow evolution.Source: §1 · p. 2
  • The paper also examines SINDy on a nine-equation turbulent shear-flow model without the CNN autoencoder.The results sections cover the wake cases and then the turbulence model.Source: §1 · p. 3

2. Methods

The methods use sparse regression to identify latent dynamics, a CNN autoencoder to compress and reconstruct flow fields, and a combined CNN-SINDy model to predict their evolution.

2.1. Sparse identification of nonlinear dynamics (SINDy)

  • SINDy regresses time derivatives against a library of candidate terms, including polynomials up to fifth order, using TLSA or adaptive Lasso.The study estimates time derivatives with a second-order central difference and obtains the coefficient matrix by regression.Source: §2.1 · pp. 3–4

2.2. Convolutional neural network-based autoencoder

  • The CNN autoencoder compresses flow data into a smaller latent space and reconstructs the input through its encoder and decoder.The latent dimension is two for the cylinder wake and its transient, and four for the two-parallel-cylinder wake.Source: §2.2 · p. 4
  • Convolutional filters and nonlinear activation support nonlinear dimension reduction, while the filter-based CNN handles high-dimensional data at lower computational cost than fully connected models.The study uses a hyperbolic tangent activation function.Source: §2.2 · p. 5

2.3. CNN-SINDy based reduced-order modeling

  • CNN-SINDy maps a flow field into latent space, fits its governing equation with SINDy, then decodes predicted latent states to reconstruct flow evolution.The authors use five-fold cross-validation to create the machine-learning models.Source: §2.3 · pp. 5–6

3. Results and discussion

For the periodic cylinder wake, Alasso yields a compact latent trajectory and CNN-SINDy reproduces flow behavior, while TLSA reveals a fit-versus-sparsity trade-off. Transient wakes can also be reproduced, but need less-sparse equations; the two-cylinder example considers a quasi-periodic wake of unequal cylinders.

3.1. Example 1: periodic cylinder wake

  • TLSA trades fit against sparsity: α = 1 gives a close but non-sparse fit, whereas α = 10 produces a parsimonious trajectory that freezes near (r1, r2) = (0.6, 0).The α = 10 decoded flow field is frozen after about t = 4.Source: §3.1 · p. 7, Figure 7 · p. 8
  • With Alasso, α = 1 × 10−4 reproduces a reference-like periodic trajectory with two terms, and CNN-SINDy reproduces the high-dimensional flow behavior.Waveform similarity is assessed by amplitude and frequency because small period differences can inflate L2 errors.Source: §3.1 · pp. 7, 9, Figure 10 · p. 11

3.2. Example 2: transient wake of cylinder flow

  • For the transient wake, TLSA gives reasonable trajectory agreement at α = 7 × 10−2, while α = 0.7 yields almost no oscillation.Source: §3.2 · p. 10
  • Alasso reproduces transient-wake trajectories and fields at low α, but higher sparsity removes developing oscillations and richer autoencoder modes accompany less-sparse equations.The authors suggest orthogonality constraints on nonlinear autoencoder modes as a possible aid to sparser dynamics.Source: §3.2 · pp. 10–11

3.3. Example 3: two-parallel cylinders wake

  • The two-cylinder test considers a quasi-periodic wake from unequal cylinders at {r, g} = {1.15, 2.0}, with DNS and reproduced fields shown at t = 60, 80, 100, and 120.Source: §3.3 · p. 12, Figure 14 · p. 13

3.4. Outlook: nine-equation shear flow model

The nine-equation shear-flow example shows that SINDy with Alasso can recover dominant terms, while its term selection becomes less reliable as training noise increases. The analysis uses a low-dimensional flow model to examine SINDy’s potential for turbulence.

  • In the nine-equation model, nine Fourier modes represent flow statistics, including mean velocity profiles, streaks, and vortex structures; SINDy is trained using 10,000 discretized coefficients.Source: §3.4 · pp. 17, 19
  • With an appropriate sparsity constant, Alasso represents the governing equations successfully, whereas TLSA produces large values and low-order overfitting.Source: Figure 21 · p. 20, §3.4 · p. 20
  • Alasso captures the dominant terms of the nine-equation turbulent shear-flow model, although coefficient magnitudes differ slightly.Source: Figure 22 · p. 20, §3.4 · p. 21
  • At γ=0.05, SINDy identifies the equation, but increasing noise removes first-order terms and adds unnecessary higher-order terms despite preserving the overall coefficient-matrix trend.Source: Figure 23 · p. 21, §3.4 · p. 21

4. Conclusion

The study applies CNN-AE and SINDy to low-dimensional representations of complex wakes and examines SINDy on a nine-equation shear-flow model. The results support equation-based modeling within these settings, while identifying a limitation of CNN-AE for flows requiring many spatial modes.

  • SINDy follows latent dynamics in the CNN-AE representation, while the coefficient-matrix term count varies with the complexity of each flow example.Source: §4 · pp. 21–22
  • For the nine-equation shear-flow model, Alasso obtains a governing equation with an appropriate sparsity constant, suggesting potential for interpretable turbulent-flow models if suitable low-dimensional mapping is available.Source: §4 · p. 22
  • The authors caution that CNN-AE may be unsuitable for flows whose energetic fields require many spatial modes, such as turbulence.Source: §4 · p. 22

Appendix A. Regression methods for SINDy

The appendix describes regression choices for identifying sparse SINDy coefficient matrices and a parameter-search procedure for selecting a sparsity constant. It contrasts thresholding and penalized regression approaches, including adaptive Lasso.

  • Linear regression estimates β by minimizing the mismatch between response P and explanatory variables Q; TLSA iteratively zeros coefficients below threshold α.Source: Appendix A · p. 23
  • Ridge constrains squared coefficient values, while Lasso penalizes their absolute values to encourage sparse models; higher α yields sparser estimates with greater error.Source: Appendix A · p. 23
  • Elastic net combines L1 and L2 penalties, while adaptive Lasso applies adaptive weights to its L1 penalty and updates weights and coefficients iteratively.Source: Appendix A · p. 24
  • The parameter search runs SINDy across candidate α values and selects a model using an evaluation index, including term counts and, for cylinder examples, amplitude and period errors.Source: Appendix A · p. 24

Appendix B. Preliminary tests with low-dimensional problems

The preliminary tests show that Alasso can recover known models over a broader range of sparsity settings than TLSA, including with enlarged libraries. Recovery nevertheless depends on suitable temporal resolution, and data preprocessing carries conditions that may constrain practical use.

Pre-test 1: van der Pol oscillator

  • Alasso identifies the van der Pol model across α = 10^-9 to 10^-2, while TLSA retains too many terms at low α.The true model has four non-zero terms; Lasso and Enet do not recover it.Source: Pre-test 1: van der Pol oscillator · pp. 25–26
  • At Δt = 0.5, neither method recovers the van der Pol model; with wider steps, Alasso retains an advantage over TLSA.At Δt = 0.2, only Alasso finds the dominant terms, while the correct-term-count range narrows as Δt increases.Source: Figure 26 · p. 26, Pre-test 1: van der Pol oscillator · p. 27
  • Training ranges longer than one period have little effect on van der Pol model identification.TLSA at α = 1 × 10^-2 and Alasso at α = 1 × 10^-3 identify the equations across all tested ranges; the Alasso result is weaker around α = 1 × 10^-8 for t = [0, 8].Source: Pre-test 1: van der Pol oscillator · p. 27, Figure 27 · p. 27
  • With libraries extended to 15th or 20th order, Alasso still identifies the van der Pol model, whereas TLSA does not.The enlarged libraries contain 136 and 231 potential terms per differential equation; near-colinear library functions produce very large TLSA coefficients.Source: Pre-test 1: van der Pol oscillator · pp. 27–28, Figure 28 · p. 28

Pre-test 2: Lorenz attractor

  • For the Lorenz system, TLSA and Alasso recover the seven-term model, but Lasso and Enet do not; Alasso succeeds across a wider α range.The comparison uses 10,000 training points over t = [0, 100] at Δt = 1 × 10^-2, with potential terms through fifth order.Source: Pre-test 2: Lorenz attractor · p. 28, Figure 30 · p. 29
  • With up to 816 potential terms per equation, Alasso identifies the Lorenz model for an appropriate α, while TLSA does not.TLSA also requires fine-tuning in this test, and Alasso coefficients are underestimated with wider-step data at Δt = 2 × 10^-2.Source: Pre-test 2: Lorenz attractor · pp. 29–30, Figure 32 · p. 30

Practical scope

  • Normalization or standardization may help Lasso and Enet, but relies on compatible training and test statistics and can add inverse-transformation error.The passage identifies ergodicity as the condition for common statistical features across training and test data.Source: Pre-test 2: Lorenz attractor · p. 30

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