Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning

DeepMDMD: Learning Algebra-Preserving Koopman Operators in Latent Space

Predicting the future motion of complex systems requires turning nonlinear physics into something mathematically manageable. Scientists often use Koopman theory to achieve this. It transforms complicated nonlinear dynamics into a linear spectral problem (a method to analyze a system using its frequencies and growth rates). In essence, instead of tracking every individual particle, you track "observables"—functions of the state that evolve according to a linear operator. However, doing this in practice is difficult. Everything depends on a hard choice: the observables must be expressive, nearly invariant under the dynamics, and compatible with the mathematical rules of composition.

Current computational methods struggle to balance these requirements. Traditional approaches use fixed "dictionaries" (sets of basis functions) that are rigid and poorly suited to complex shapes. Other methods use deep learning to find flexible coordinates. However, these often ignore the underlying algebraic laws of the system. This creates a tension between flexibility and structure. A new paper introduces DeepMDMD (Deep Embedded Multiplicative Dynamic Mode Decomposition). This method resolves this conflict. It teaches a computer to find a simplified "hidden" version of the system while strictly enforcing the fundamental mathematical rules that govern it.

The failure of geometric partitioning

The core challenge in approximating the Koopman operator lies in the choice of the "dictionary." In Multiplicative Dynamic Mode Decomposition (MDMD), a technique that enforces the Koopman product rule, the dictionary is composed of indicator functions. These act like "on/off" switches. They indicate if a system is within a specific region or cell of the state space.

Because these functions are tied to specific regions, the problem becomes one of partitioning the state space into cells. Researchers have traditionally relied on k-means clustering to create these partitions. While robust, k-means is fundamentally geometric. It looks for clusters based on spatial distance rather than how the system actually evolves. As shown in, a standard k-means partition for a nonlinear pendulum cuts across trajectories.

This creates cell boundaries that do not respect the underlying physics. In high-dimensional systems, this "geometry-first" approach suffers from the curse of dimensionality. It requires massive dictionaries to compensate for poorly aligned boundaries. This misalignment leads to "spectral pollution," which refers to the appearance of spurious, non-physical eigenvalues that corrupt predictions.

Learning coordinates while constraining algebra

DeepMDMD breaks the dependence on ambient geometry. It moves the partitioning process from the high-dimensional physical state space into a low-dimensional, learned latent space. The methodology follows an alternating optimization pipeline .

1. Autoencoder Pretraining: The process begins by training an autoencoder. This is a neural network that uses an encoder to compress data and a decoder to reconstruct it. This establishes a low-dimensional latent representation ($Z$) of the original state space ($X$).
2. Exact Operator Update: With a fixed partition in the latent space, the algorithm computes the Koopman matrix ($K$) using the MDMD algorithm. This step is not a mere approximation. It enforces the multiplicative structure as a hard algebraic constraint. This ensures the nonzero spectrum remains on the unit circle. This prevents the model from predicting physically impossible exponential growth or decay.
3. Differentiable Partition Update: To refine the cells, the authors employ a "soft" relaxation of the cluster assignments. Hard assignments are not differentiable. This means they cannot be optimized via gradient descent (a common method for training neural networks). The authors use a Student t-kernel (a heavy-tailed probability distribution) to allow each latent point to contribute to multiple cells during training .

This allows the encoder and the cluster centroids to be updated via backpropagation. This effectively "shapes" the latent space so that the cells align with the actual dynamics.

By alternating these steps, the latent space becomes a coordinate system sculpted to promote "Koopman closure." This is the condition where the chosen functions stay within their own span under the action of the operator.

Superior spectral resolution and robustness

The authors demonstrate that this dual approach provides advantages in accuracy and efficiency. In a benchmark test of a nonlinear pendulum, the paper reports that DeepMDMD achieves comparable accuracy to standard MDMD. However, it does so using roughly an order of magnitude fewer basis functions .

This efficiency is vital for scaling to larger problems.

The method also excels at capturing the "continuous spectrum." This refers to the complex, frequency-dependent behaviors found in many physical systems. The authors measure the "rank" of a log-modulus matrix to quantify how many algebraically independent eigenfunctions the model resolves. As illustrated in, DeepMDMD produces a significantly slower decay in singular values compared to MDMD or EDMD.

This means it resolves a much richer and more accurate set of spectral structures.

The strength of the method is most evident in high-dimensional, noisy environments. In tests involving a 158,624-dimensional cylinder wake, DeepMDMD's latent-space forecasting acted as a natural denoiser. The forecast is performed entirely within the compressed latent bottleneck. Noise that does not conform to the learned manifold is filtered out. The paper reports that even with 40% Gaussian noise, DeepMDMD maintains stable, coherent trajectories. In contrast, traditional state-space MDMD fails to maintain long-term spectral statistics [, Figure 10].

Limitations of the alternating scheme

Despite these successes, the DeepMDMD framework has clear trade-offs. First, the method relies on an alternating optimization scheme. This involves switching between the operator update and the partition update. The authors note that this scheme lacks a formal theoretical proof of convergence. Users may encounter sensitivity to the starting conditions of the autoencoder.

Second, the training process is computationally intensive. Standard DMD is often a single-shot linear regression. DeepMDMD requires multiple epochs of gradient descent. It also requires repeated re-computation of the Koopman matrix. This "front-loads" the computational cost. While the resulting model is extremely fast for long-term forecasting, the initial discovery phase is demanding.

Finally, the choice of the reconstruction weight ($\lambda$) is currently empirical. This weight balances the importance of reconstructing the original data against modeling the dynamics. There is no automated way provided to tune this parameter. Practitioners must rely on manual experimentation to find the optimal balance.

The verdict: A new rule for Koopman learning

DeepMDMD represents a shift in how we approach data-driven dynamical systems. It suggests that the most effective way to model complex physics is to use deep learning and algebraic constraints together. By "learning the coordinates" through an autoencoder and "constraining the algebra" through MDMD, the method produces models that are compact and spectrally rich.

For engineers working with high-dimensional fluid dynamics or noisy sensor data, this is a practical tool. The ability to perform stable, long-term rollouts in a denoised latent space offers a path toward reliable digital twins. The code is available for reproduction at https://github.com/kelangray/DeepMDMD.

Figures from the paper