Deep Learning Based Surrogate Modeling of PDE Governed Systems Using Fourier Neural Operators (FNOs): Application to Clarifier Dynamics in Wastewater Treatment

Mihirkumar Patel, Shankar B. Kausley, Shirish Karande

Keywords: Fourier Neural Operator (FNO), Clarifier Modeling, Wastewater Treatment, Surrogate Modeling, PDE, Digital Twin, Deep Learning

Abstract: Clarifiers are essential units in wastewater treatment, enabling the sedimentation of suspended solids and ensuring effluent quality. Traditional modeling of clarifier dynamics relies on solving nonlinear partial differential equations (PDEs) that incorporate advection, diffusion, and settling velocity terms. While accurate, these numerical solvers are computationally intensive and unsuitable for real-time applications such as process control, optimization, or digital twin integration. On the other hand, empirical and rule-based models offer faster computations but often lack robustness and fail to capture the complex, nonlinear behavior of sludge settling under diverse operational scenarios. To overcome these limitations, this study introduces a data-driven surrogate modeling framework using the Fourier Neural Operator (FNO), a deep learning architecture capable of learning solution operators for PDE governed systems to simulate clarifier dynamics efficiently and accurately. A validated one-dimensional clarifier simulator was developed to generate synthetic datasets under varying operational conditions, including different initial concentration distributions, flow rates, and feed concentrations. The FNO model was trained to map initial and boundary conditions to final concentration profiles, effectively capturing the complex, nonlinear dynamics of sludge settling. The trained model achieved relative L2 errors below 1 \%, demonstrated robust generalization to unseen scenarios, and delivered inference speeds over 1000× faster than conventional (Finite Difference Method) solver. This approach enables the creation of accurate, scalable digital twins for clarifiers, supporting rapid scenario analysis and real-time decision-making. Limitations include minor boundary inaccuracies and fixed time horizon predictions, which may be addressed through hybrid modeling approaches like Physics-Informed Neural Operator (PINO). This study highlights the effectiveness of FNOs as fast, scalable, and mesh-independent surrogate models for systems governed by PDEs, offering a practical foundation for their integration into real-time simulation, control, and digital twin technologies in process engineering.

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Submission Number: 73