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Physics-Informed Neural Modeling of TiO 2 Photoanode Thickness-Dependent Electron Transport and Photovoltaic Performance in Dye-Sensitized Solar Cells

This study presents a physics-informed neural network framework that accurately models the thickness-dependent electron transport and photovoltaic performance of dye-sensitized solar cells using limited experimental data, outperforming traditional regression methods while highlighting the challenges of cross-system generalization due to morphological and electrolyte variations.

Original authors: K Sanyasi Naidu, Nalluri Mohan Rao

Published 2026-09-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: K Sanyasi Naidu, Nalluri Mohan Rao

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Solar energy has long promised a future where sunlight is converted directly into electricity, but the path to that future is paved with microscopic challenges. One of the most promising ways to capture light involves a device called a dye-sensitized solar cell. Imagine a sponge made of tiny, porous ceramic particles, soaked in a light-sensitive dye, and sandwiched between two electrodes. When sunlight hits this sponge, the dye releases electrons, which then travel through the sponge's network to generate power. The thickness of this ceramic sponge is a critical design choice. If the layer is too thin, it cannot absorb enough light to create a strong current. If it is too thick, the electrons get lost or trapped before they can reach the exit, wasting the energy they carried. Finding the perfect balance between capturing light and transporting electrons is a delicate act of engineering, one that researchers have been trying to solve by testing different layer thicknesses in the lab.

In a recent study, researchers K. Sanyasi Naidu and Nalluri Mohan Rao from Jawaharlal Nehru Technological University in India tackled this problem by combining the laws of physics with a new kind of computer modeling. Instead of relying solely on trial-and-error experiments or simple statistical guesses, they built a digital model that respects the fundamental rules of how electrons move through the material. They focused on a specific set of six solar cells, each made with a different thickness of titanium dioxide, a common ceramic material used in these devices. The goal was to create a tool that could predict how changing the thickness would affect the cell's performance, even when the data available was very sparse.

The researchers used a technique known as a physics-informed neural network. To understand this, think of it as teaching a computer to solve a puzzle where the rules are already known. In this case, the rules are the physical equations that describe how electrons diffuse through the material and how they recombine with other particles, effectively disappearing. The computer was not just given the experimental results to memorize; it was forced to learn a solution that obeyed these physical laws. The model treated the thickness of the solar cell as a variable that could be adjusted, allowing the researchers to simulate what would happen at any thickness between the thinnest and thinnest samples they had tested. This approach allowed them to see the underlying trends that simple data fitting might miss.

The team started with data from six specific devices, with thicknesses ranging from 12.7 micrometers to 55.2 micrometers. They found that as the layer got thicker, the amount of light captured increased, but so did the chance that electrons would get lost. The model successfully identified a sweet spot where these competing effects balanced out. It predicted that the most efficient thickness for this specific type of solar cell would be around 29.72 micrometers, capable of converting about 8.70% of the sunlight into electricity. This prediction was slightly higher than the best thickness they had actually tested, which was 26.6 micrometers, suggesting that the true optimum might lie just beyond their experimental range.

However, the researchers were careful not to overstate their findings. They tested their model rigorously by pretending they didn't have data for one of the six devices at a time, then checking if the model could accurately predict the performance of that missing device. The model performed significantly better than traditional mathematical methods, such as simple curve fitting or standard artificial intelligence models that ignore the laws of physics. While the model could predict the performance of the known devices with reasonable accuracy, the researchers noted that the small amount of data meant there was still a wide range of uncertainty about exactly where the perfect thickness lies. The "best" thickness could reasonably be anywhere between roughly 13 and 55 micrometers, highlighting that more data is needed to pinpoint the exact number.

Perhaps the most revealing part of the study came when the researchers tried to use their model on a completely different set of solar cells made by other scientists using different materials and manufacturing methods. When they applied their "frozen" model to these new devices without making any adjustments, the predictions were not accurate. The model failed to predict the correct optimal thickness for the new system. This failure was not a mistake, but a crucial discovery. It demonstrated that there is no single, universal rule for how thickness affects solar cell performance across all different types of devices. The specific way the material is made, the type of dye used, and the chemical environment all change the rules of the game.

Ultimately, this study offers a powerful new way to think about solar cell design. It shows that by embedding the laws of physics directly into computer models, researchers can make better predictions even when they have very little data. The model successfully captured the complex trade-off between light absorption and electron transport for the specific devices it was trained on. Yet, it also clearly showed the limits of trying to generalize these findings. The thickness of the ceramic layer is not the only factor that matters; the entire manufacturing process and material composition must be considered to truly optimize a solar cell. This work provides a reproducible framework for combining sparse experimental data with physical laws, offering a more reliable path forward for designing efficient solar energy systems than relying on guesswork or simple data trends alone.

Technical Summary: Physics-Informed Neural Modeling of TiO₂ Photoanode Thickness-Dependent Electron Transport in DSSCs

Problem Statement
In Dye-Sensitized Solar Cells (DSSCs), the thickness of the TiO₂ photoanode is a critical, non-monotonic variable. Increasing thickness enhances dye loading and optical absorption but simultaneously increases the electron transport distance, leading to higher recombination rates and transport losses. While traditional polynomial or spline interpolations can approximate thickness trends, they fail to enforce the underlying electron continuity equation. Conversely, deterministic Partial Differential Equation (PDE) models often require effective coefficients that are difficult to uniquely identify from sparse experimental data. The study addresses the challenge of modeling this thickness-dependent performance with a limited dataset (six devices) while distinguishing between within-system predictive capability and universal generalization.

Methodology
The authors developed a Physics-Informed Neural Network (PINN) framework based on a one-dimensional steady-state electron continuity equation. The approach integrates physical laws with a data-driven residual correction to handle sparse data.

Mathematical Formulation:
- The model solves a normalized diffusion-generation-recombination equation for the excess electron density $u(\xi, d)$ , where $\xi$ is the normalized depth and $d$ is the film thickness.
- The PDE explicitly incorporates thickness via the dimensionless groups $(L/d)^2$ (diffusion vs. recombination) and $\alpha d$ (optical attenuation).
- Boundary conditions include ideal extraction at the FTO contact ( $u=0$ ) and zero flux at the electrolyte interface.
- Key physical parameters (effective electron diffusion coefficient $D_e$ , lifetime $\tau$ , absorption coefficient $\alpha$ , and photon flux $\phi$ ) were fixed to specific effective values to avoid non-identifiability issues inherent in fitting six data points with multiple free parameters.
Neural Architecture:
- The solution field $u_\theta$ is approximated using a Physics-Informed Extreme Learning Machine (PI-ELM) with random features.
- The network uses a single hidden layer with 80 tanh features; weights and biases are frozen, and only the output coefficients are optimized via a constrained least-squares problem.
- This architecture avoids back-propagation costs and ensures analytical differentiability for PDE residuals.
Device-Level Calibration:
- To bridge the gap between the idealized 1D PDE and real device performance, a low-capacity regularized residual correction was applied.
- The short-circuit current ( $J_{SC}$ ) was modeled as the PDE-derived current multiplied by a residual term: $\exp(c_0 + c_1 z + c_2 z^2)$ , where $z$ is the normalized thickness. The coefficients $c_1$ and $c_2$ were constrained by L2 ridge regularization ( $\lambda=0.10$ ) to prevent overfitting.
- $V_{OC}$ and Fill Factor (FF) were modeled using linear residual mappings.
Validation Protocol:
- Analytical Verification: The neural field was compared against the analytical solution of the constant-coefficient PDE.
- Leave-One-Thickness-Out Cross-Validation (LOOCV): The model was trained on five devices and tested on the sixth, repeated for all six samples.
- Baseline Comparison: Performance was benchmarked against linear/quadratic regression, natural cubic splines, Gaussian Process Regression, and a conventional Artificial Neural Network (ANN).
- External Transfer: A "frozen" model (trained on the Tsai et al. dataset) was applied without recalibration to two independent published datasets (Sedghi & Miankushki; Kumari et al.) to test domain shift.

Key Results

Physics-Field Accuracy: The neural field solved the PDE with high fidelity, achieving a relative L2 error of 0.08–0.68% against the analytical solution across all thicknesses.
Predictive Performance (LOOCV):
- The PINN achieved a Mean Absolute Percentage Error (MAPE) of 12.40% for efficiency, significantly outperforming all baselines: Linear Regression (27.76%), Quadratic Regression (26.08%), Natural Cubic Spline (78.42%), Gaussian Process (19.34%), and Conventional ANN (20.15%).
- The model predicted an optimum thickness of 29.72 µm with an efficiency of 8.70%, slightly higher than the experimental maximum at 26.6 µm, reflecting the model's robustness in the high-performance plateau.
Ablation Studies:
- Removing the physics constraint (data-only ANN) increased the efficiency MAPE to 20.15%.
- Using only the physics model with a global current scale (no residual correction) yielded a MAPE of 12.35%, demonstrating that the PDE shape captures the majority of the trend, while the residual correction refines the fit.
Uncertainty Quantification: Due to the small dataset (N=6), a jackknife resampling analysis revealed a wide uncertainty interval for the optimum thickness (12.72–46.72 µm), indicating that the precise optimum is not statistically resolvable with only six data points.
External Transfer Limitations:
- When applied to the Sedghi & Miankushki dataset, the frozen model yielded an efficiency MAPE of 17.10%.
- When applied to the Kumari et al. dataset (different fabrication, dye, and electrolyte), the MAPE increased to 19.78%.
- The model failed to capture the lower optimum thickness of the external systems, confirming that thickness alone is insufficient to describe performance across different fabrication routes.

Significance and Claims
The paper claims to present a physics-regularized small-data framework that successfully discriminates between within-system predictive value and universal generalization.

Reproducibility and Robustness: The study demonstrates that embedding transport physics into a neural model allows for superior interpolation and extrapolation within a specific system compared to purely data-driven methods, even with very sparse data (N=6).
Modesty in Generalization: The authors explicitly state that the goal is not to predict a universal DSSC performance curve. Instead, the study highlights the "domain shift" problem: a thickness-only surrogate cannot generalize across different fabrication chemistries (e.g., compression, sintering, dye choice).
Methodological Shift: The work advocates for a conservative modeling approach where physical constants are fixed to avoid non-identifiability, and empirical corrections are strictly regularized. This prevents the model from "memorizing" noise in small datasets while maintaining physical consistency.
Future Direction: The paper concludes that for broader applicability, future models must evolve into multivariate frameworks incorporating explicit inputs such as porosity, dye loading, and transport resistance, rather than relying solely on thickness as a proxy.

Technical Summary: Physics-Informed Neural Modeling of TiO₂ Photoanode Thickness-Dependent Electron Transport in DSSCs

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