1. Cover Email (Gmail body)

To: info@impulsespace.com
Subject: Robertson stiff ODE -- 2,336x gain in PINN training
Attachments: PINN_Training_Breakthrough_Stefan_Tender.pdf, pinn_academic_benchmarks.png, pinn_fluid_physics_benchmarks.png

To the Impulse Space team -- I would appreciate this being forwarded to Mr. Mueller. It concerns a technical result in neural PDE solvers relevant to propulsion and combustion modeling. Mr. Mueller, I built a training modification for Physics-Informed Neural Networks that resolves multi-scale gradient conflict in stiff systems. Robertson ODE (stiffness 7.5 x 10^8): standard PINN gives 65.4% error. My method reduces it to 0.03%. Same network, same optimizer, same seed -- the only change is how gradients are handled during training. Tested across 6 standard benchmarks. Geometric mean improvement: 480x. The harder the problem, the larger the gain. Given your experience with combustion chemistry stiffness, I think you will immediately see what this means. I attached a one-page summary with benchmarks and setup. If this looks interesting, I can walk you through it in 10 minutes. - Stefan Tender bruce.tender@gmail.com

2. PDF Attachment

Solving the Stiff ODE Problem in Neural PDE Solvers

Stefan Tender | March 2026 | bruce.tender@gmail.com

Why I Am Writing to You Specifically

You spent two decades solving the hardest propulsion engineering problems on the planet. The combustion chemistry behind Merlin and Raptor involves stiff ODE systems with rate constants spanning 8+ orders of magnitude -- the kind of problem that breaks most numerical methods.

I built something that solves exactly that class of problem using neural networks. And you are one of very few people who would immediately understand what the numbers below actually mean.

The Problem You Know

Physics-Informed Neural Networks (PINNs) should be able to learn stiff ODE systems directly. In practice, they fail catastrophically -- the competing loss terms from fast and slow timescales create gradient conflicts that the optimizer cannot resolve.

Every PINN paper says "future work" when it hits stiffness ratios above 10^4. I solved it.

The Numbers

Same network, same optimizer, same epochs, same seed. Only difference: my training modification. No architecture changes, no hyperparameters, < 3% computational overhead.

Stiff & Propulsion-Relevant Problems

Problem	Standard PINN	My Method	Gain
Robertson Stiff ODE (stiffness 7.5 x 10^8)	65.4%	0.03%	2,336x
Burgers Equation (viscous shock, Re~1050)	15.8%	0.55%	28.7x
Helmholtz Equation (k^2=60)	67.8%	0.77%	87.8x

Robertson's rate constants: 0.04, 10^4, and 3 x 10^7. Standard PINNs produce output worse than a trivial guess (65.4% error). My method resolves all three species to 0.03%.

Additional Benchmarks

Problem	Standard PINN	My Method	Gain
Poisson (k=5)	556%	0.22%	2,483x
Poisson (k=10)	282%	0.04%	6,883x
2D Diffusion	33.2%	0.27%	122x

Geometric mean across all 6: 480x. The harder the problem, the larger the gain. Not tuned -- the method adapts on its own.

What This Means in Your World

The Robertson stiffness ratio of 7.5 x 10^8 is comparable to what you see in hydrocarbon combustion mechanisms -- the chemistry behind every engine you have ever built. If PINNs could handle that stiffness natively, combustion modeling shifts from days of classical simulation to real-time inference.

The Burgers shock at Re~1050 captures the advection-dominated dynamics relevant to nozzle flow. Standard PINNs smear the shock entirely. My method resolves it to 0.55%.

I am not claiming this replaces Cantera tomorrow. I am saying the fundamental obstacle -- multi-scale gradient conflict in neural PDE training -- is solved. The applications follow from there.

Methodology & Setup

Hardware: NVIDIA RTX PRO 6000 (96 GB VRAM), 255 GB RAM, Python 3.12, PyTorch CUDA.

Pure PyTorch implementation -- no PINN library dependencies
Deterministic seeds (seed=42) -- every number reproducible on any machine with CUDA
Network: 6 hidden layers x 512 neurons (1.3M parameters) -- same architecture for ALL problems
Optimizer: Adam, lr=10^-3, 5,000 epochs -- same for Standard and Enhanced
The ONLY difference between Standard and Enhanced is the training modification. Zero tuning per problem
Reference solutions: Cole-Hopf transform (Burgers), SciPy BDF at rtol=10^-10 (Robertson), closed-form analytical (Poisson, Diffusion, Helmholtz)
All 6 problems shown -- no cherry-picking, no excluded runs
Results stable across seeds: tested with seeds 0-9, gains remain within same order of magnitude

Patent pending. Full technical details available under NDA.

What I have shown here is the most straightforward application of a deeper principle. The same core idea produces results in domains that have nothing to do with PDEs.

I built this alone. What I need now is a technical partner with the infrastructure to bring it to the real world. What I have shown here is significant -- but what I have not shown here is bigger.

bruce.tender@gmail.com