UKF ↔ PINN fusion (covariance intersection)¶
The UKF and the PINN family have complementary strengths: the UKF is well-posed and well-calibrated on the pH / charge-balance states, the PINN (smoother or amortised observer) is strong on the biogas-driving states and forecasts naturally. The fusion combines their two state trajectories so the result keeps the best of each.
1. The idea: Covariance Intersection¶
We have two estimates of the same state, but we do not know how correlated their errors are (they see the same plant and the same noise). Covariance Intersection (CI) fuses them conservatively without assuming independence, so the fused estimate stays consistent (never over-confident) and is at least as informative as either input.
Per state, CI weights by information (inverse variance): the more confident estimate contributes more. A single weight \(\omega \in [0,1]\) per time step trades the two off.
2. The mathematics¶
Take two diagonal-covariance estimates with means \(m_a, m_b\) and per-state variances \(v_a = \sigma_a^2,\ v_b = \sigma_b^2\), and write the information (inverse variance) as \(I = 1/v\). CI fuses them as
with fused variance \(v_f = 1/I_f\). The weight \(\omega\) is either fixed or, per time step, optimised by a grid search that minimises the fused covariance determinant. Equivalently, it maximises \(\sum_\text{states}\log I_f\).
Because \(\omega\) is chosen per step and the covariances are diagonal (built from the per-state standard deviations), the fusion is cheap and applies to a whole trajectory at once.
3. Implementation¶
fuse_ci_diagonal(mean_a, std_a, mean_b, std_b, omega=None) is the core: a
per-time-step CI of two (T, n) estimates, returning the fused mean, fused std,
and the chosen \(\omega\) per step.
HybridEstimator wraps it in the BatchEstimator contract. Build it
from the two sub-estimates, and estimate() returns their CI fusion as a
TrajectoryEstimate:
from pyadm1ode_estimation.estimation.fusion import HybridEstimator
hybrid = HybridEstimator(traj_ukf, traj_pinn) # two TrajectoryEstimates
fused = hybrid.estimate() # CI fusion on the shared grid
Both inputs must live on the same time grid, fuse_trajectories_ci enforces
matching shapes.
4. Caveat: calibration is a prerequisite¶
CI weights purely by covariance, so it is only as good as the inputs' calibration. An over-confident estimator, e.g. MC-Dropout bands that are too narrow, is over-trusted and drags the fusion toward it. In the twin comparison the fusion never hurts, but it cannot yet blend the UKF's pH strength into the state estimate, because the PINN's uncertainty is not yet calibrated. Calibrating the PINN band (and fixing the pH conditioning) is the open item that unlocks the fusion.
Source files¶
pyadm1ode_estimation/estimation/fusion/hybrid.py—HybridEstimator,fuse_ci_diagonal,fuse_trajectories_ci
API reference¶
pyadm1ode_estimation.estimation.fusion.hybrid.HybridEstimator(traj_a, traj_b, omega=None, var_floor=1e-12)
¶
Variant C: fuses two trajectory estimates (e.g. UKF + PINN) via CI.
Conforms to the :class:~pyadm1ode_estimation.estimation.base.BatchEstimator
output contract. Build it with the two sub-estimates already computed over
the same window; :meth:estimate returns their CI fusion.
pyadm1ode_estimation.estimation.fusion.hybrid.fuse_ci_diagonal(mean_a, std_a, mean_b, std_b, omega=None, n_grid=21, var_floor=1e-12)
¶
Per-time-step Covariance Intersection of two diagonal-covariance estimates.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
mean_a, std_a, mean_b, std_b
|
|
required | |
omega
|
float | None
|
fixed fusion weight in |
None
|
n_grid
|
int
|
grid resolution for the ω search. |
21
|
var_floor
|
float
|
added to variances for numerical safety (zero-std inputs). |
1e-12
|
Returns:
| Type | Description |
|---|---|
ndarray
|
|
ndarray
|
|