Modeling hysteresis in stage-discharge: Physics and Artificial Intelligence based Approach

The stage-discharge rating curve is crucial for flow estimation in open channels. The power relationship between discharge (Q) and stage (h) is used conventionally to evaluate the discharge from stage measurements. However, this relationship performs poorly under unsteady conditions, varying bed roughness and alteration of cross-sectional geometry. Hysteretic behavior is often found in the rating curves showing different discharges under identical stages because of the unsteadiness. Attempts have been made by the scientific community to develop such Q-h relationship that can be able to capture this hysteresis along with easy computations. Jones' formula comprises steady state discharge and temporal gradient of h is one of the equations that have been used for modeling this hysteresis. Symbolic regression (SR) has also been applied to trained machine learning (ML) models to derive site-specific explicit mathematical Q-h equation of high accuracy. However, the SR-based relationship does not exhibit the realistic hysteretic nature of rating curves. This study aims to find a robust stage-discharge relationship that shall capture the realistic hysteretic nature while having high accuracy. To achieve this, a Physics-Informed Neural Network (PINN) is developed incorporating the Jones formula into its loss function along with the data-driven error term and a term to calibrate the parameters of the Jones formula. Further, SR is implemented using the PySR module to derive a mathematical equation that fits the prediction of the PINN. This equation has no differential term and incorporates the stage on time t, stage on time (t-1) and steady state discharge. For the Q-h data with 15-minute temporal resolution of the River Brays of the USA, the rating curves derived from the Jones formula and this PINN-SR are compared based on their abilities to capture the hysteretic nature of the Q-h relationship. Four metrics of hysteresis capturing performance and an overall score are used for comparison. All data are normalized to avoid mixed units in the overall score. The hysteresis area error to check the magnitude is found to be 1.574 for Jones formula and 0.129 for PINN-SR. For fitting accuracy, the average of Root Mean Square Errors (RMSEs) for rising and falling limbs are 0.425 and 0.360, the hysteresis width errors are 0.007 and 0.138, and the Direction-Aware Dynamic Time Warpings (DTWs) are 18.225 and 5.752. The overall error scores for hysteresis are 5.058 and 1.595 for the Jones formula and PINN-SR-based rating curves, respectively. These results indicate the superior performance of PINN-SR-based rating curve over the Jones formula in capturing the hysteresis under unsteady flow conditions.