Lilan: A linear latent network approach for real-time solutions of stiff, nonlinear, ordinary differential equations (2501.08423v3)

Abstract: Solving stiff ordinary differential equations (StODEs) requires sophisticated numerical solvers, which are often computationally expensive. In particular, StODE's often cannot be solved with traditional explicit time integration schemes and one must resort to costly implicit methods to compute solutions. On the other hand, state-of-the-art ML based methods such as Neural ODE (NODE) poorly handle the timescale separation of various elements of the solutions to StODEs and require expensive implicit solvers for integration at inference time. In this work, we embark on a different path which involves learning a latent dynamics for StODEs, in which one completely avoids numerical integration. To that end, we consider a constant velocity latent dynamical system whose solution is a sequence of straight lines. Given the initial condition and parameters of the ODE, the encoder networks learn the slope (i.e the constant velocity) and the initial condition for the latent dynamics. In other words, the solution of the original dynamics is encoded into a sequence of straight lines which can be decoded back to retrieve the actual solution as and when required. Another key idea in our approach is a nonlinear transformation of time, which allows for the "stretching/squeezing" of time in the latent space, thereby allowing for varying levels of attention to different temporal regions in the solution. Additionally, we provide a simple universal-approximation-type proof showing that our approach can approximate the solution of stiff nonlinear system on a compact set to any degree of accuracy, {\epsilon}. We show that the dimension of the latent dynamical system in our approach is independent of {\epsilon}. Numerical investigation on prototype StODEs suggest that our method outperforms state-of-the art machine learning approaches for handling StODEs.

• The paper introduces a latent dynamics model that transforms stiff ODEs into linear trajectories in a latent space, eliminating the need for numerical integration.
• The paper employs a nonlinear time transformation to effectively capture varying temporal dynamics, achieving computational speedups of up to three orders of magnitude.
• The paper demonstrates that encoder-decoder architectures can approximate stiff system solutions with strong generalization and high accuracy.

An Analytical Overview of a Latent Dynamics Approach for Stiff ODEs

The paper under discussion presents a novel approach to addressing the computational challenges associated with solving stiff ordinary differential equations (StODEs). These equations are prevalent in various scientific and engineering fields, and their inherent numerical stiffness demands sophisticated solvers that are often computationally intensive. Traditional explicit integration methods are inadequate due to the stiff nature of these systems, necessitating the use of implicit methods, which can be significantly more costly in terms of computational resources. The paper introduces a machine learning-based technique that leverages neural networks to avoid the need for numerical integration altogether by learning a latent dynamics model.

Key Methodological Innovations

The paper proposes a latent dynamical system characterized by constant velocity, where solutions manifest as sequences of straight lines in a transformed latent space. This approach negates the requirement for numerical integration within the latent space, establishing a computationally efficient surrogate model for the original stiff dynamics.

1. Latent Dynamics Model: The model posits a latent space where the equation solutions are represented as simple linear trajectories. By employing encoder neural networks, the initial conditions and parameters of the differential equations are transformed into initial conditions and velocities for these latent trajectories.
2. Nonlinear Time Transformation: A significant aspect of this methodology is the nonlinear transformation of the time variable, which allows the model to 'stretch' or 'squeeze' time in the latent space. This feature enables varying levels of attention across different temporal regions of the solution, catering to the dynamics of stiff systems which operate across multiple timescales.
3. Universal Approximation Capability: The authors provide a theoretical grounding for their approach, showing that their latent model can approximate the solutions of stiff systems to any desired accuracy. Moreover, it is highlighted that the latent space's dimensionality is independent of the accuracy level, conferring scalability to high-dimensional problems.
4. Encoder and Decoder Networks: The implementation consists of encoder networks that map the initial condition and the system parameters into the latent space, and decoder networks that map back from the latent space to the original system dynamics.

Validation and Results

The authors validate their approach on benchmark problems, including the Robertson stiff chemical kinetics model and a high-dimensional collisional-radiative (CR) model. These experiments demonstrate the proposed method's ability to outperform state-of-the-art machine learning techniques, such as deep operator networks (DeepONet) and Neural ODEs, both in terms of computational efficiency and accuracy.

• Performance Metrics: The proposed method achieved notable reductions in error metrics compared to competing approaches, offering significant computational speedup of up to three orders of magnitude over traditional stiff ODE solvers.
• Generalization and Overfitting Prevention: The method shows strong generalization capabilities, avoiding the overfitting pitfalls that commonly afflict neural network-based surrogates when trained on limited data.

Implications and Future Directions

The implications of this research are broad, offering a viable pathway for efficiently solving stiff ODEs without the computational burden of standard numerical solvers. This development holds promise for real-time systems and applications where high efficiency is paramount.

The authors suggest future research directions, including a deeper theoretical exploration of why their method generalizes better than direct flow map learning and adaptive data sampling to reduce training data requirements. These advancements could further enhance the method's applicability to a wider range of complex dynamical systems.

In summary, the paper presents a compelling method to circumvent the computational challenges posed by stiff ODEs, leveraging latent space dynamics and machine learning innovations. This marks a significant stride in surrogate modeling for computationally demanding tasks.