Evaluation of Essential AI’s RNJ-1 for Differential Equations
Essential AI’s RNJ-1 is an 8-billion-parameter open-weight model based on a Gemma-3 style architecture with global attention and an extended context window. It is trained on more than eight trillion tokens and tuned for coding, mathematics, and general STEM reasoning.
my interest is specifically in its mathematical ability, I tested it directly on a sequence of differential equations, increasing the difficulty step by step.
The five equations I used were:
dy/dx – y = e^x
x * dy/dx + 2y = x^3
y'' – 4y' + 5y = e^(2x) * (x + 1)
y'' – 4y' + 4y = e^(2x) * (x + 1)
x^2 * y'' – 3x * y' + 4y = x^3
RNJ-1 solved four of these five equations correctly. It handled both first-order equations cleanly, produced the correct complementary and particular solutions for the non-resonant second-order equation, and solved the Cauchy–Euler equation exactly as expected. The only failure was on equation (4), the resonant second-order case where the forcing term overlaps with the homogeneous solution. In that problem, the model chose the wrong trial form for the particular solution and did not include the additional power of x required to break the resonance.
Aside from that one miss, RNJ-1’s reasoning was clear and consistent. It followed standard methods, including integrating factors, characteristic equations, and trial-particular functions, and it carried out the algebra correctly. The only practical issue I encountered was that very large context lengths in ZeroGPU caused truncation or runtime errors; at normal context lengths, the model behaved stably.
Compared with other models in this size class that I have used for solving similat differential equations— RNJ-1 performs noticeably better on routine STEM tasks. Its derivations are cleaner, and its structure is more reliable. it produced output in latex form but did not render it, while deepseek math renders the math equation.
Other than its difficulty with the resonance case and the occasional instability at very large context sizes, RNJ-1 is one of the strongest open-source 8B models available today for solving differential equations.
I was not able to run its gguf , quantized version on mh phone. it needs a recompile of llama.cpp
The model is running using zerogpu at this Space
https://huggingface.co/spaces/Javedalam/EssentialAI
