CNN1D monotone-integral model architecture

Build the physics-informed 1-D CNN with a monotone integral output layer, as described in Norouzi et al. (2025).

Details

Architecture

The model takes two inputs:

1. Xseq_knots: a 3-D tensor of shape [N, K, p+1] — for each observation, p scaled covariates are broadcast across K knot positions, and the knot positions themselves form the last channel.

2. pf_norm: a 2-D tensor of shape [N, 1] — the query pF value normalised to [0, 1].

The output satisfies: $$\hat{\theta}(pF) = \theta_s - \int_0^{pF} \text{softplus}(s(t))\,dt$$

where \(s(t)\) is a 1-channel convolutional output. Monotone decrease is guaranteed by construction because the integrand is always positive.

References

Norouzi, A. M., et al. (2025). Physics-Informed Neural Networks for Estimating a Continuous Form of the Soil Water Retention Curve. Journal of Hydrology.