Path-minimizing Latent ODEs for improved extrapolation and inference (2410.08923v1)

Abstract: Latent ODE models provide flexible descriptions of dynamic systems, but they can struggle with extrapolation and predicting complicated non-linear dynamics. The latent ODE approach implicitly relies on encoders to identify unknown system parameters and initial conditions, whereas the evaluation times are known and directly provided to the ODE solver. This dichotomy can be exploited by encouraging time-independent latent representations. By replacing the common variational penalty in latent space with an $\ell_2$ penalty on the path length of each system, the models learn data representations that can easily be distinguished from those of systems with different configurations. This results in faster training, smaller models, more accurate interpolation and long-time extrapolation compared to the baseline ODE models with GRU, RNN, and LSTM encoder/decoders on tests with damped harmonic oscillator, self-gravitating fluid, and predator-prey systems. We also demonstrate superior results for simulation-based inference of the Lotka-Volterra parameters and initial conditions by using the latents as data summaries for a conditional normalizing flow. Our change to the training loss is agnostic to the specific recognition network used by the decoder and can therefore easily be adopted by other latent ODE models.

• The paper introduces a novel ℓ2 path-length penalty in latent ODEs to replace the conventional KL penalty, leading to improved extrapolation and inference.
• The method leverages differential geometry to enforce structured latent space, significantly reducing reconstruction errors in dynamic systems.
• Experimental results on models like the damped oscillator and predator-prey equations demonstrate robust performance and enhanced parameter recovery.

Overview of Path-Minimizing Latent ODEs for Improved Extrapolation and Inference

The paper presents a novel approach to training Latent Ordinary Differential Equations (ODEs) focused on enhancing their interpolation, extrapolation, and inference capabilities. By introducing a path-length penalty in latent space, the authors propose a methodological enhancement that allows for more accurate modeling of dynamic systems, presenting an improvement over traditional latent ODE models which employ a variational penalty.

Problem Context

Latent ODEs have been widely used for modeling sequential data due to their flexibility in capturing dynamic systems. However, challenges remain in extrapolating long-term dependencies and parameter inference, often due to issues like the vanishing gradient problem in RNN-based encoders. Traditional methods using variational autoencoders (VAEs) can struggle in accurately depicting complex systems over time.

Methodological Contributions

The key idea is to replace the conventional variational KL penalty with an $\ell_2$ path-length penalty. This change encourages time-independent latent representations, focusing on minimizing trajectory path lengths in latent space. This regularization effectively segments latent space according to unknown parameters and initial conditions of the system, yielding more structured latent encodings.

The method leverages concepts from differential geometry, akin to finding geodesics, to promote short paths in latent space, reducing temporal variation. This approach can be adopted with any recognition network without architecture modifications, highlighting its adaptability.

Experimental Validation

The authors validate their approach across three test cases:

1. Damped Harmonic Oscillator: The proposed model outperformed the baseline in both interpolation and extrapolation tasks. It demonstrated significantly reduced reconstruction errors and shorter path lengths in latent space, leading to better inference of initial conditions.
2. Self-Gravitating Fluid (Lane-Emden Equation): In this test, the baseline models struggled, whereas the path-minimized model yielded accurate solutions even for out-of-distribution data. The paper attributes this success to a more effective encoding that inherently captures discrete system states.
3. Lotka-Volterra Equations (Predator-Prey Model): The model effectively structured latent space around system parameters, facilitating improved extrapolation and robust inference even with scarce data, outperforming both baseline approaches and alternative training methods like HBNODE.

Implications and Future Work

The proposed path-minimizing strategy showcases significant potential for improving the stability and accuracy of latent ODE models. By focusing on accurately capturing system parameters in latent space, the approach enhances both forecasting and inference capabilities. The introduction of this technique suggests a shift toward more geometry-inspired methods in neural sequence modeling.

Future research could explore combining this approach with modular neural ODE architectures to delineate static and dynamic states explicitly. Moreover, examining chaotic system behavior and extending this regularization to stochastic differential equations presents further research avenues.

Conclusion

The paper provides a substantial contribution to the field of sequence modeling with latent ODEs. By redefining the loss landscape, it offers a compelling path to more robust and interpretable models, capable of handling complex dynamic systems with greater fidelity. This work opens the door for more nuanced applications in simulation-based inference and invites exploration of more geometric frameworks in neural ODEs.