## 1 Introduction  
Cheese whey is a waste generated in cheese production, which is characterized by its high organic load [1]. Different biological and physicochemical treatments have been proposed to recover compounds with added value, such as proteins and lactose. However, a significant number of small and medium industries do not have the necessary economic resources to implement technologies for the treatment of whey, which leads to the inadequate disposal of the waste, causing serious problems of environmental pollution [2]. Anaerobic treatment is an alternative for the valorization of this effluent. The anaerobic digestion of cheese whey reduces the organic load with simultaneous recovery of value-added by-products such as VFA, methane-rich biogas, biohydrogen, and biopolymers [3, 4]. Anaerobic digestion (AD) is a complex and intrinsically unstable process. Indeed, the anaerobic treatment is formed by consecutive stages conducted by a specific group of microorganisms with different metabolic characteristics which are not fully understood. Moreover, variations of the input variables (hydraulic flow rate, influent organic load) and the effects of inhibitory agents may easily lead to the wash-out condition [5]. Thus, proper diagnosis and control of AD are of utmost relevance to avoid serious breakdowns and guarantee operational stability [6; 7]. To this end, it is necessary to monitor the variables that provide information on the performance of each stage. The key variables of each stage can be defined by their relationship with the performance of the associated microbial consortium. Indeed, carbohydrate and protein degradation are related to hydrolysis, VFA production with acidogenesis-acetylogenesis, and biogas production with acetogenesis-methanogenesis [8]. However, at the industrial level, in situ and in-line measurements in AD plants are limited to pH, temperature, water flow, biogas flow, level, and pressure [9; 10; 11]. Although pH and biogas production can be used as indicators of the overall status of the AD process, they fail to indicate the stress level of the reactor in real time [12]. On the one hand, biogas production can show the overall process performance. However, a decrease in biogas production often occurred after the process was severely inhibited. On the other hand, pH is not a suitable parameter for a well-buffered bioreactor since pH has low sensitivity. VFA concentration has been suggested as an ideal indicator [12]. Indeed, VFA accumulation is usually interpreted as the result of methanogenesis inhibition or organic overloading, leading to a risk of process breakdown. Thus, for AD process diagnosis, several authors have proposed different thresholds based on VFA measurements as early warning indicators to judge the reactor status [5; 13; 14; 15; 16]. Hence, for the AD diagnosis, different model-based and data-driven estimation approaches have been proposed to determine key variables, including VFA concentration, using the available information in real time (e.g., temperature, pH, conductivity measurements). Model-based estimation methods are based on the reconstruction of unmeasured variables from the measurements available through the mathematical model of the process [11; 17]. Model-based estimation approaches applied to AD processes include state-observers [18; 19; 20; 21; 22], Bayesian methods [23], and filtering techniques [24; 25]. The most significant limitation of model-based approaches is that they depend on the accuracy of the process model, i.e., the lower the precision of the model, the greater the estimation error. Moreover, model uncertainties degrade model predictions, and the measurements are affected by noise and disturbances. Data-driven methods are based on statistical models trained to learn trends from collected historical [26].