RJ-PINNs: Residual Jacobian Physics-Informed Neural Networks for Guaranteed Convergence

RJ-PINNs, introduced by Dadoyi Dadesso, is a pioneering framework using Jacobian-based least-squares (TRF) to directly minimize residuals without traditional loss functions. This method is the first to eliminate gradient optimizers in PINNs, offering unmatched robustness for inverse PDE problems.

For more information, please refer to the following:https://github.com/dadesso17/RJ-PINNs/ or preprint: https://doi.org/10.5281/zenodo.15138086

Features

• Jacobian-based least-squares (TRF): Directly minimize residuals without traditional loss functions.
• No Gradient Optimizers: Eliminate the need for gradient optimizers in PINNs.
• Robustness for Inverse PDE Problems: Unmatched robustness for solving inverse partial differential equations (PDE) problems.

Key Differences

Aspect	Traditional PINNs	RJ-PINNs
Objective	Minimize a loss function L(θ)	Minimize the residuals R(θ)
Gradient	Compute ∇L(θ)	Compute ∇R(θ)
Optimization	Use gradient-based optimizers (e.g., Adam, L-BFGS)	Use least-squares optimizer (e.g., TRF)
Implementation	Define a loss function and its gradient	Define residuals and their Jacobian
Convergence	Not guaranteed	Robust convergence

Table: Comparison between Traditional PINNs and RJ-PINNs

RJ-PINNs Diagram

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## ⚠️ Important: Practical Notes for Using RJ-PINNs

RJ-PINNs (Residual Jacobian Physics-Informed Neural Networks) offer better convergence and stability than traditional PINNs — but some practical issues can still arise.

🚨 When Problems Occur

• In inverse problems involving multiple parameter identification, or in some complex direct problems, RJ-PINNs may still diverge.
• This is often caused by a rapid decrease of the physics residual R_physics, leading to instability or divergence.

✅ How to Fix It

• For direct problems without observed data:

  • 🔧 Decrease the weight w_p applied to R_physics (e.g., 1e-3).
  • • For direct problems with observed data:
  • 🔼 Increase the weight w_d on R_data (e.g., 1e4,).
  • 🔽 Decrease the weight w_p on R_physics (e.g., set to 1e-1...).

• For inverse problems:

  • 🔼 Increase the weight w_d on R_data (e.g., 1e4,).
  • 🔽 Decrease the weight w_p on R_physics (e.g., set to 1e-1).

• 🧠 Alternative strategies:

  • Use adaptive weighting techniques or normalization strategies (as in traditional PINNs) to improve stability.

🧩 Note: This issue is common in the general PINN framework — it's not specific to RJ-PINNs.

Citation

If you use RJ-PINNs in your research, please cite:

@software{Dadesso_RJ-PINNs_2025,
  author = {Dadesso, D.},
  year = {(2025)},

  title = {{Residual Jacobian Physics-Informed Neural Networks (RJ-PINNs) for Guaranteed Convergence}},
  version = {1.0},
  publisher = {Zenodo},
  doi = { 10.5281/zenodo.15138086},

}