PINN is predicting trivial solution for stiff ODE [D]

2026-05-15 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Advanced, quick

Summary

A user is encountering difficulties applying Physics-Informed Neural Networks (PINNs) to solve a second-order ordinary differential equation (ODE) representing a damped harmonic oscillator. The specific ODE is m*d2y/dt2 + mu*dy/dt + k*y = 0, with boundary conditions y(t=0) = 1 and y'(t=0) = 0. The PINN model successfully predicts solutions for stiffness parameter 'k' values up to 50, but yields trivial solutions when 'k' exceeds this threshold. Attempts to resolve the issue, including reducing the learning rate, increasing data points, reusing weights, and incrementally increasing 'k' in smaller steps, have been unsuccessful. The problem suggests a limitation of standard PINN approaches for stiff ODEs.

Key takeaway

For Machine Learning Engineers developing PINN models for differential equations, if your model yields trivial solutions for stiff ODEs, you should consider implementing second-order optimizers or curriculum learning. These techniques can help the network converge to non-trivial solutions by better navigating complex loss landscapes, especially when dealing with high stiffness parameters.

Key insights

PINNs struggle with stiff ODEs, often predicting trivial solutions beyond certain stiffness thresholds.

Principles

• Stiff ODEs challenge standard PINN training.
• Second-order optimizers can improve PINN performance.

Method

Employ second-order optimizers like Gauss-Newton or Self-scale Quasi-Newton. Alternatively, implement curriculum learning by gradually increasing ODE stiffness during training.

In practice

• Use second-order optimizers for stiff problems.
• Implement curriculum learning for complex ODEs.
• Explore physics-informed kernel learning.

Topics

• Physics-Informed Neural Networks
• Stiff ODEs
• Damped Harmonic Oscillator
• Second-Order Optimizers
• Curriculum Learning

Best for: AI Student, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.