Sharp Stability Threshold and Certification for Designing Stable Residual Architectures

We propose \emph{the sublinear-growth principle} for deep residual architectures -- a sharp stability threshold on the input-magnitude exponent of every residual block's velocity field: $$\|v(x, t)\| \leq c\,\|x\|^q + b, \qquad q \in [0, 1].$$ The threshold $q = 1$ is established via two independent arguments. Classical ODE theory gives a global forward flow on $[0, T]$ at $q \le 1$ and exhibits divergent velocity fields at any $q > 1$. The optimal-control analysis, via the Hamilton-Jacobi-Bellman equation, sharpens this to a selection statement: the training optimum is bang-bang on the boundary of the admissible class, so the optimum at $q > 1$ blows up while the optimum at $q \le 1$ is safe by construction. The exponent criterion $q \le 1$ is thereby a necessary and sufficient condition for stable training. It clarifies architectural placements that ensure the stability of training and inference, explaining, for instance, the stabilizing role of layer normalization. The sublinear-growth velocity fields form \emph{the right function space} on which forward dynamics, adjoint sensitivity, and architectural composition are all well-controlled. An arithmetic of input-magnitude exponents under the five operations that build residual blocks enables efficient certification of $q_k \le 1$ at the level of architectural primitives, in place of ad hoc trial and error in the search for stable neural architectural designs. A parameter-free modification reduces the supercritical Mamba block from $q = 5$ to $q = 1$ without layer normalization, demonstrating this point. Experiments on Mamba and PatchTST confirm that the $q \le 1$ variants train stably: the criterion is the input-magnitude exponent, not the presence of a normalization layer.

ISOMORPH: A Supply Chain Digital Twin for Simulation, Dataset Generation, and Forecasting Benchmarks

Open time-series forecasting (TSF) benchmarks cover retail, energy, weather, and traffic, but supply-chain logistics remains underserved. We introduce ISOMORPH, the first public digital twin of a multi-echelon logistics network with fully interpretable, user-configurable parameters and modular topology, demand process, and control rules. The simulator advances a directed routing graph in discrete time: demand arrives at the destination, is served from stock or recorded as backlog, and triggers replenishment through the network. The state vector tracks per-node on-hand inventory with outstanding orders, in-transit shipments, and a smoothed demand estimate, so the dynamics close as a Markov chain on a tractable state space whose transition kernel acts linearly on the empirical distribution of the state. The released data reproduces the bullwhip effect at empirically consistent magnitudes, and three conservation laws encoded in the Markov chain serve as verification tools when users extend the simulator. We release datasets at two catalogue scales ($C=50$ and $C=200$) with six scenario sweeps producing 30 additional rollouts and 20 Latin-hypercube perturbations, exhibiting dynamics absent from fixed TSF benchmarks: variance amplification, cascading bottlenecks, regime shifts, and cross-channel coupling through shared macro shocks. Zero-shot evaluation of four foundation models (Chronos, Moirai, TimesFM, Lag-Llama) shows MASE values exceeding public GIFT-Eval references at low-to-moderate horizons, supporting incorporation into existing benchmarks. The same pairing produces forecast confidence bands via Latin-hypercube perturbation of demand-side knobs, forward UQ from parameter uncertainty unavailable on standard TSF datasets, demonstrating that foundation models can serve as fast surrogates for the digital twin's forward UQ. Code (MIT): https://github.com/tuhinsahai/ISOMORPH.