Invariant NODEs

I.A. Auzina, C. Yıldız, S. Magliacane, M. Bethge and E. Gavves

Invariant NODEs

Key Idea:

length, mass

 length of limbs

influence dynamics

influence reconstruction

 color of a ball

color of clothes

does setting apart dynamic states from

underlying static factors of variation improve existing model performance?

Invariant NODEs

Key Idea:

does setting apart dynamic states from

underlying static factors of variation improve existing model performance?

influence dynamics

influence reconstruction

Figure 1.  Schematic illustration of our method.

Invariant NODEs

Figure 1.  Schematic illustration of our method.

Modulated NODEs

I.A. Auzina, C. Yıldız, S. Magliacane, M. Bethge and E. Gavves

Modulated NODEs

I.A. Auzina, C. Yıldız, S. Magliacane, M. Bethge and E. Gavves

\mathbf{d} \sim p(\mathbf{d}) ~\quad \textit{// dynamics modulator} \\
\mathbf{s} \sim p(\mathbf{s}) ~~\quad \textit{// static modulator} \\
\mathbf{z}_1 \sim p(\mathbf{z}_1) \quad \textit{// latent ODE state} \\

I. Generative Model

\mathbf{x}_i \sim p_{\mathbf{x}_i}(\mathbf{x}_i \mid \mathbf{z}_i ~;~ \mathbf{s}).
\mathbf{z}_i = \mathbf{z}_1 + \int_{t_1}^{t_i} \mathbf{f}_{\mathbf{\theta}}(\mathbf{z}(\tau);\mathbf{d})~d \tau \\

Modulated NODEs

I.A. Auzina, C. Yıldız, S. Magliacane, M. Bethge and E. Gavves

\mathbf{d} \sim p(\mathbf{d}) ~\quad \textit{// dynamics modulator} \\
\mathbf{s} \sim p(\mathbf{s}) ~~\quad \textit{// static modulator} \\

II. Learning Latent Modulator variables

(i) static modulator

\mathbf{s}^{n} = \frac{1}{T} \sum_{i=1}^{T} \mathbf{s}^{n}_i, \qquad \mathbf{s}^{n}_i = \mathbf{g}_\psi\big(\mathbf{x}^{n}_i\big).

// compute average over the observation embedings

time-invariance

// concatenate with latent ODE state

\mathbf{x}^{n}_i \sim p_{\mathbf{x}_i}\Big(\mathbf{x}^{n}_i \Big\vert \big[\mathbf{z}^{n}_i,\mathbf{s}^{n}\big]\Big) ~~ \forall i \in [1,\ldots,T].

Modulated NODEs

I.A. Auzina, C. Yıldız, S. Magliacane, M. Bethge and E. Gavves

\mathbf{d} \sim p(\mathbf{d}) ~\quad \textit{// dynamics modulator} \\
\mathbf{s} \sim p(\mathbf{s}) ~~\quad \textit{// static modulator} \\

II. Learning Latent Modulator variables

(i) dynamics modulator

\mathbf{d}^{n} = \frac{1}{T-T_e} \sum_{i=1}^{T-T_e} \mathbf{d}^{n}_i, \qquad \mathbf{d}^{n}_i = \mathbf{h}_{\upsilon}\big(\mathbf{x}^{n}_{i:i+T_e}\big),

// compute over subsequences of length

time-invariance

// pass as input to the differential function

\frac{\textrm{d}\mathbf{z}^{n}(t)}{\textrm{d}t} = \mathbf{f}_\mathbf{\theta} \big(\mathbf{z}^{n}(t), \mathbf{d}^{n} \big)
T_e

Modulated NODEs

I.A. Auzina, C. Yıldız, S. Magliacane, M. Bethge and E. Gavves

A general framework

Latent NODE

(Chen et al., 2018)

Second Order NODE

(Norchliffe et al., 2020)

Latent Second Order NODE

(Yildiz et al., 2019)

Heavy Ball NODE

(Xia et al., 2021)

can be applied to most x-NODE

Modulated NODEs

I.A. Auzina, C. Yıldız, S. Magliacane, M. Bethge and E. Gavves

A general framework

that improves performance

Sinusoidal Data

Latent NODE

(Chen et al., 2018)

Modulated NODEs

I.A. Auzina, C. Yıldız, S. Magliacane, M. Bethge and E. Gavves

A general framework

that improves performance

Predator-Prey (PP) data

Heavy Ball NODE

(Xia et al., 2021)

Modulated NODEs

I.A. Auzina, C. Yıldız, S. Magliacane, M. Bethge and E. Gavves

that improves performance

with respect to forecasting and generalization

with respect to disentanglement

Table 1. Test MSE and its standard deviation with and without our framework. Lower is better.

Table 2. R^2 scroes to predict the unknown Factors of Variation from inferred latents. Higher is better.

Modulated NODEs

I.A. Auzina, C. Yıldız, S. Magliacane, M. Bethge and E. Gavves

that improves performance

on real world data

Table 3. Test MSE and standard deviation. Lower is better.

MoNODE

NODE

Modulated NODEs

I.A. Auzina, C. Yıldız, S. Magliacane, M. Bethge and E. Gavves

We introduce a modulator framework for NODE models that allows to separate time-evolving ODE states from modulator variables

Limitations/Extensions

  • express constants as modulator variables

Benefits

  • improved generalization
  • improved far-horizon forecasting
  • capture true known FoV
  • limited to deterministic systems
  • out of distribution modulators
  • self-supervised contrastive loss
  • interpretability /disentanglement

Modulated ....

I.A. Auzina, C. Yıldız, S. Magliacane, M. Bethge and E. Gavves

Biology Applications

Object-centric representations

Thank you for your attention

I.A. Auzina, C. Yıldız, S. Magliacane, M. Bethge and E. Gavves

Modulated NODEs

Modulated NODEs Boston University 2023

By iaa

Modulated NODEs Boston University 2023

  • 205