Hybrid hydrologcial models

Concepts in H2M

Parameters in hybrid models are categorized into ML parameters and physical parameters¹. Thus, hybrid modeling are grouped into parameteric (both learned) and non-parameteric (the latter parameters fixed) ones.

H2M structure

The end-to-end hybrid hydrological model

Creative points

• A physical state $s_{t-1}$ is taken into the recurrent neural network $ g_{RNN} $ for accounting memory effects.

\[h_{t}=g_{RNN}(h_{t-1},[x_{t},s_{t-1}])\]

• A mapping output function $g_{out}$ that links certain latent variable or coefficient $p_{t}$ in process-based models with $h_{t}$

• The process-based model $f_{pb}$ considers not only original physical constraints $s_{t}$, but also memory effects from RNN.

\[y_{t}, s_{t}=f_{pb}(p_{t},x_{t},s_{t-1})\]

Comparisons with others

1. H2M
   • Pros
     • hard physical constraints
     • additional insights of latent variables and coefficients
     • partial interpretability
   • Cons
     • qunatification of uncertainties
     • Generalizability needs to be investigated in certain cases
2. Regularization via loss functions (soft constriants)
   • penalizing physically inconsistent results
   • Pros
     • additional means in diagosing physical inconsistency
   • Cons
     • penalizing physically inconsistent results does not always make sense
     • lacks of hard physical constraints
     • no insights via latent variables and coefficients
3. Mass conserving neural networks (hard constraints)
   • adding inductive biase
   • Pros
     • enhances robustness and generalizability
     • good performance in extreme events
   • Cons
     • does not outperform non-mass conserving architecture
     • limited interpretability
4. Data assimilation
   • combining simulations from process-based models and obervation for the optimal estimates of geophysical states
   • Pros
     • aim to quantify errors
   • Cons
     • remaining model errors

Personal Opinions

• Compared to physics-constrained machine learning², using neural networks to replace empirical parameter, it steps further in linking RNNs with latent variables (coefficients) in process-based models, which further taps the potential of AI.
• A question is why acknowledge latent variables (coefficients) activated by softplus function after LSTM layer as Evapotranspration and snow water equvilatent in multi-task layerhttps://ieeexplore.ieee.org/document/8578879, and how to figure out its effect.

Footnote

1. Deep learning and hybrid modeling of global vegetation and hydrology. Basil Kraft. 2022. ↩

2. Zhao, W. L., Gentine, P., Reichstein, M., Zhang, Y., Zhou, S., Wen, Y., et al. (2019). Physics-constrained machine learning of evapotranspiration. Geophysical Research Letters, 46, 14496–14507. https://doi.org/10.1029/2019GL085291 ↩