Simple physical systems like neurons may be fundamentally impossible for digital computers to simulate efficiently, no matter how much we scale.
April 15, 2026
Original Paper
Complexity Theory meets Ordinary Differential Equations
arXiv · 2604.09790
The Takeaway
By applying complexity theory to Ordinary Differential Equations (ODEs), this paper proves a 'complexity blowup' in analog information processing. For most linear ODEs, low-complexity inputs can lead to outputs that require more-than-polynomial computation time to approximate. This suggests a fundamental limit on our ability to simulate biological brains or complex physical systems on digital hardware. It challenges the vision of 'whole brain emulation' and suggests we might need analog or neuromorphic hardware to truly mimic nature's efficiency. Digital simulation is hitting a mathematical wall.
From the abstract
This contribution investigates the computational complexity of simulating linear ordinary differential equations (ODEs) on digital computers. We provide an exact characterization of the complexity blowup for a class of ODEs of arbitrary order based on their algebraic properties, extending previous characterization of first order ODEs. Complexity blowup indeed arises in most ODEs (except for certain degenerate cases) and means that there exists a low complexity input signal, which can be generate