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On the Identifiability of Hybrid Deep Generative Models: Meta-Learning as a Solution
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Shows that hybrid deep generative models with unconditional priors are unidentifiable and proposes a meta-learning formulation (conditional prior on the neural latent given few-shot context samples) that yields provable identifiability, validated on pendulum, advection-diffusion and double-pendulum data.
Problem
Hybrid deep generative models that combine physics-based expressions with neural components cannot be identified from data: the neural component is theoretically unidentifiable and can overpower the physics parameters, so identifiability has not been established.
Approach
Replace the unconditional prior on the neural latent with a conditional exponential-family prior whose parameters are computed from a set of few-shot context samples drawn from the same data-generating process, so the meta-hybrid-DGM satisfies nonlinear-ICA-style conditional identifiability without observed auxiliary labels.
Key results
- On the forced damped pendulum, Meta-Hybrid-VAE reached MCC 0.99 for the neural latent versus 0.59 for Hybrid-VAE and 0.79 for APHYNITY, with prediction MSE 6.85e-2 versus 1.74 and 2.37.
- MCC exceeded 0.85 once 7 training tasks were used for the pendulum, matching the theoretical threshold of de+1=7 for d=3, e=2.
- On the real double pendulum, meta-hybrid-VAE kept prediction MSE near reconstruction level while baselines degraded by 2-3 orders of magnitude.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use meta-learning with context-conditioned priors rather than an unconditional prior | Hybrid-DGM · Forced Damped Pendulum | MCC: 0.59(0.03) -> 0.99(0.00) | Helped |
| use meta-learning with context-conditioned priors rather than an unconditional prior | Hybrid-DGM · Advection-Diffusion System | MCC: 0.94(0.00) -> 0.99(0.00) | Helped |
| use meta-learning with context-conditioned priors rather than an unconditional prior | Hybrid-DGM · Double Pendulum | MCC: 0.51(0.00) -> 0.98(0.00) | Helped |
| use meta-learning with context-conditioned priors rather than an unconditional prior | Hybrid-DGM · Forced Damped Pendulum | MSE of x (Rec): 2.66(0.03)e-2 -> 2.91(0.06)e-2 | Mixed |
| use meta-learning with context-conditioned priors rather than an unconditional prior | Hybrid-DGM · Real Double Pendulum | MSE of x (Pre): 1.87 -> 2.88(0.34)e-2 | Helped |
| vary the number of unknown parameters to 2 instead of 4 (ablation) | Meta-Hybrid-VAE · Forced Damped Pendulum | MCC: 0.99(0.00) -> 0.97(0.01) | No clear effect |
| vary the number of unknown parameters to 3 instead of 4 (ablation) | Meta-Hybrid-VAE · Forced Damped Pendulum | MCC: 0.99(0.00) -> 0.99(0.00) | No clear effect |
| remove meta-conditioning of zP (ablation) | Meta-Hybrid-VAE · Forced Damped Pendulum | MSE of zP: 5.63(0.42)e-3 -> 9.17(1.33)e-3 | Hurt |
| remove meta-conditioning of zP (ablation) | Meta-Hybrid-VAE · Forced Damped Pendulum | MSE of x (Pre): 3.37(0.00)e-2 -> 8.82(2.03)e-2 | Hurt |
| increase the number of training tasks to at least 7 (ablation) | Meta-Hybrid-VAE · Forced Damped Pendulum | MCC | Helped |
Surprising findings
- The meta-formulation's prediction MSE on the real double pendulum was much lower than baselines even though reconstruction MSE was comparable across all models, showing reconstruction alone does not reveal identifiability.
Limitations
- Requires the ability to pair context and query samples from the same data-generating process; not applicable when no pairing information is available.
- Ablations on context set size and neural latent dimension are not yet comprehensive.
- Compute resources and training cost are not reported; broader impact is not discussed.
Experimental setup
Datasets: forced damped pendulum (synthetic) advection-diffusion system (synthetic) double pendulum (synthetic) real double pendulum videos (Hybrid-DGM dataset of [4])
Benchmarks: forced damped pendulum advection-diffusion system double pendulum real double pendulum
Metrics: MSE of zP MCC MSE of x reconstruction MSE of x prediction strong MCC
Baselines: APHYNITY Hybrid-VAE (physics-integrated hybrid VAE) iVAE Non-Meta zP
Builds on: iVAE APHYNITY physics-integrated hybrid VAE nonlinear ICA
Closest papers by meaning
- Position: Understanding LLMs Requires More Than Statistical Generalization (ICML 2024 · similarity 0.87)
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