To establish a reference trajectory, the researchers used an explicit Sumudu transform–Adomian decomposition recursion. In an independent recomputation, the magnitude of component 50 on the 1001-point grid was 3.43018×10⁻²⁰, and the maximum residual under extended-precision evaluation was approximately 1.69974×10⁻⁹. A product-trapezoidal solver exhibited second-order convergence, providing a benchmark for evaluating the neural approaches.

The experiments used a chronological split applied before sequence windowing. Across five random seeds, the teacher-forced LSTM achieved a test charge RMSE of 0.01537 ± 0.00369, while autonomous rollout gave 0.16095 ± 0.04806. The fPINN, with the fractional residual enforced over the whole interval, reached a test RMSE of 2.15109×10⁻⁴ ± 2.03608×10⁻⁴.

By contrast, a network penalized by an integer-order residual — the wrong order of physics for this fractional system — yielded 0.05414 ± 1.84137×10⁻⁴. The three feedforward variants trained without any residual on the test interval had mean test RMSEs ranging from 0.39211 to 0.92453, more than three orders of magnitude above the fPINN result.

The authors emphasize that these differences reflect the information and prior knowledge supplied to each model rather than an intrinsic ranking of architectures. In this comparison, supplying the correct fractional-order residual in the training objective was associated with the lowest errors on the test interval.

The study’s scope is limited. The data were synthetic and noise-free, only one parameter set of the circuit was studied, and the extrapolation interval was short. The reported results therefore characterize performance under these specific conditions rather than establishing general superiority for all fractional-order modeling tasks.

Within those bounds, the quantitative contrast is clear: the fPINN’s mean test RMSE of roughly 2×10⁻⁴ was far below the 0.05414 of the integer-order residual network, the 0.01537 of the teacher-forced LSTM, and the 0.39211 to 0.92453 range of the data-only feedforward variants. The autonomous LSTM rollout, at 0.16095 ± 0.04806, also fell well short of the physics-guided result.

The work provides a controlled, like-for-like comparison of data-driven and physics-guided neural approaches on a fractional-order circuit problem, with all models evaluated against the same reference trajectory and error metric. The reported numbers offer a concrete baseline for future studies of neural modeling of fractional-order systems.

Subject of Research: Comparative evaluation of data-driven and physics-guided neural networks for modeling a fractional-order LCR circuit

Article Title: Data-driven and physics-guided neural approaches for LCR circuit modeling: a comparative analysis

Article References: Gopi Sai Ram, T., & Sarkar, N. (2026). Data-driven and physics-guided neural approaches for LCR circuit modeling: a comparative analysis. Scientific Reports. https://doi.org/10.1038/s41598-026-74713-y

Image Credits: AI Generated

DOI: 10.1038/s41598-026-74713-y

Keywords: fPINN, LSTM, physics-informed neural networks, fractional calculus, LCR circuit, Caputo derivative, scientific machine learning, RMSE, Adomian decomposition, Sumudu transform, Scientific Reports, Data-driven