PhysicsInformed Neural Networks for Dynamic Process Operations with Limited Physical Knowledge and Data
Core Concepts
Physicsinformed neural networks (PINNs) can estimate unmeasured process states even when respective constitutive equations are unknown, provided that the available physical knowledge is sufficient.
Abstract
The content explores the use of physicsinformed neural networks (PINNs) for modeling dynamic processes with incomplete mechanistic semiexplicit differentialalgebraic equation systems and scarce process data. The key insights are:

PINNs can infer immeasurable states with reasonable accuracy, even if respective constitutive equations are unknown. This is demonstrated through numerical examples of a continuously stirred tank reactor (CSTR) and a liquidliquid separator.

The authors propose an incidence matrixbased heuristic to assess whether estimation of unmeasured states may be possible with a PINN. The heuristic mimics structural index analysis and indicates the potential for state estimation based on the available physical knowledge.

For the CSTR example, the authors show that the PINN models outperform a purely datadriven vanilla neural network benchmark in predicting the measured states, especially in the lowdata regime. The PINN models can also accurately estimate the unmeasured algebraic states, even when their constitutive equations are not provided.

For the liquidliquid separator example, the authors demonstrate that the PINN model can estimate unmeasured differential states with good accuracy when the available physical knowledge is sufficient, as indicated by the incidence matrix analysis.
The authors conclude that PINNs constitute a promising approach for modeling dynamic processes when relatively few experimental data and only partially known mechanistic descriptions are available.
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PhysicsInformed Neural Networks for Dynamic Process Operations with Limited Physical Knowledge and Data
Stats
The reactor volume V(t) and the coolant flow rate QK(t) are the manipulated variables.
The concentrations cA(t) and cB(t), the reactor temperature T(t), and the cooling jacket temperature TK(t) are the process states.
Quotes
"PINNs can infer immeasurable states with reasonable accuracy, even if respective constitutive equations are unknown."
"The authors propose an incidence matrixbased heuristic to assess whether estimation of unmeasured states may be possible with a PINN."
"The PINN models can also accurately estimate the unmeasured algebraic states, even when their constitutive equations are not provided."
Deeper Inquiries
How can the proposed PINN approach be extended to handle more complex dynamic processes with higherdimensional state spaces and more involved physical models?
The proposed PhysicsInformed Neural Network (PINN) approach can be extended to handle more complex dynamic processes by incorporating several strategies. First, the architecture of the neural network can be enhanced to accommodate higherdimensional state spaces. This can be achieved by increasing the depth and width of the neural network, allowing it to learn more intricate relationships among the inputs and outputs. Additionally, advanced techniques such as convolutional layers or recurrent layers can be integrated to capture spatial and temporal dependencies more effectively.
Second, the incorporation of multifidelity modeling can be beneficial. By combining lowfidelity models (which may be computationally cheaper but less accurate) with highfidelity models (which are more accurate but computationally expensive), the PINN can leverage the strengths of both to improve predictions in complex systems. This hybrid approach can also facilitate the training process by providing a richer dataset.
Third, the extension of the PINN framework to include more sophisticated physical models can be achieved by integrating additional governing equations that describe the dynamics of the system. This may involve using more complex differentialalgebraic equation (DAE) systems or even partial differential equations (PDEs) that account for spatial variations in the process. Furthermore, the PINN can be adapted to include multiscale modeling, where different scales of the process (e.g., molecular, mesoscopic, and macroscopic) are considered simultaneously.
Lastly, the use of transfer learning can be explored, where a PINN trained on a simpler or related problem can be finetuned for a more complex problem. This approach can significantly reduce the amount of data required for training and improve the model's generalization capabilities in higherdimensional state spaces.
What are the limitations of the incidence matrixbased heuristic, and how can it be improved to provide more reliable assessments of PINN state estimation capabilities?
The incidence matrixbased heuristic, while useful for assessing the potential for state estimation in PINNs, has several limitations. One significant limitation is that it provides a binary assessment of the presence or absence of relationships between unmeasured states and the governing equations. This simplification may overlook the nuances of how these relationships manifest in practice, leading to false positives or negatives regarding the feasibility of state estimation.
Additionally, the heuristic does not account for the quality or quantity of available data, which can significantly impact the PINN's ability to estimate states accurately. For instance, even if the incidence matrix indicates that state estimation is possible, insufficient or noisy data can still lead to poor performance.
To improve the reliability of the incidence matrixbased heuristic, it could be enhanced by incorporating quantitative measures of data quality and availability. This could involve integrating metrics such as data density, noise levels, and the correlation between measured and unmeasured states into the analysis. Furthermore, a probabilistic approach could be adopted, where the heuristic provides a likelihood score for successful state estimation based on the incidence matrix and data characteristics.
Another improvement could involve the use of sensitivity analysis to evaluate how variations in the measured states affect the unmeasured states. By understanding the sensitivity of the system, practitioners can better assess the robustness of the state estimation capabilities indicated by the incidence matrix.
What other applications in chemical engineering or beyond could benefit from the PINN modeling approach presented in this work?
The PINN modeling approach has a wide range of potential applications in chemical engineering and beyond. In chemical engineering, it can be utilized for realtime process monitoring and control, where the ability to estimate unmeasured states can enhance the operational efficiency of reactors, separators, and other unit operations. For instance, in batch processes, PINNs can help optimize conditions by predicting concentrations and temperatures that are difficult to measure directly.
Beyond chemical engineering, the PINN approach can be applied in environmental engineering for modeling pollutant dispersion in air and water bodies, where the governing equations are often complex and data may be scarce. In biomedical engineering, PINNs can assist in modeling biological processes, such as drug delivery systems or metabolic pathways, where understanding unmeasured states is crucial for effective treatment strategies.
Moreover, the PINN framework can be extended to fields such as finance, where it can model complex systems with partial knowledge of underlying dynamics, or in robotics, where it can be used for state estimation in dynamic environments with limited sensor data. The versatility of PINNs makes them a promising tool for tackling a variety of challenges across different domains, particularly where traditional modeling approaches struggle due to incomplete data or complex physical phenomena.