MiNO: Cotangent-bundle propagator learning for PDEs

Scientific machine learning for partial differential equations commonly targets solution fields, as in physics-informed neural networks, or solution maps, as in neural operators. We study a third target: the propagator itself, a phase and amplitude in phase space. The motivation is a gap in regularity. A transported discontinuity is nonsmooth in space and time, yet the rule that moves it can be a polynomial phase carrying unit amplitude, so the object that generates an evolution can be far smoother than the field it generates. The microlocal neural operator (MiNO) learns that object, using the eikonal equation for the phase and the transport equation for the amplitude, and recovers the solution by an oscillatory integral. Sharp fronts and caustics then belong to propagation geometry rather than to a field fitted pointwise. Small residuals certify more than the reconstructed field. They place the learned canonical relation, the geometry that carries singularities, close to the exact one, and they separate trainable error from the frequency-truncation tail. On a matched-budget discontinuous-advection benchmark, MiNO stops improving within 10,000 steps at the accuracy limit of its finite reconstruction window, a limit predicted in closed form, whereas a physics-informed neural network with neural-tangent-kernel loss balancing stays near its initial error. On smooth advection, the mean error is $3.84\times10^{-3}$ for MiNO and $3.12\times10^{-2}$ for a supervised Fourier neural operator. Single-branch MiNO is the smallest model compared, and one trained generator serves five unseen initial conditions without retraining.

SEAM: Global consistency beyond local accuracy in scientific machine learning

Scientific machine learning commonly validates models at the level of a subdomain, a benchmark split, or an explanation for one prediction. Yet such local checks cannot establish whether the resulting explanations can be assembled into one globally admissible explanation. We introduce Scientific Explanation-Admissibility Machines (SEAM), a generator-agnostic framework that makes this local-to-global consistency question computable across regions, sensors, regimes, and model components. The finite explanation-sheaf instantiation SEAM-$Ω$ represents each region by a structured explanation with state, closure, and observation channels together with optional contract metadata; compares neighboring explanations on their overlaps; and converts disagreement into a channel-resolved obstruction. This obstruction locates inconsistency and tests competing declared accounts by restricting each repair to the revisions that one account permits. Exact feasibility refutes or retains an account; when exact repair is unavailable, residual-aware regularized records provide a separately labeled empirical attribution. The framework also separates inconsistency from non-identifiability and monitors learned generators under distribution shift. We establish theorems for minimum-cost intervention and conservation-contract detectability, together with companion results for identifiability and closure recoverability. Across nineteen experiments involving synthetic partial differential equation systems and out-of-distribution Fourier neural operator (FNO) monitoring, SEAM detects incompatible explanations even when local predictions are accurate, and attributes failures to specific channels and overlaps. SEAM adds a global explanation-consistency audit to existing solvers and learning models, testing whether their local explanations form a coherent scientific account.

Looped Transformers with Source-Centered State Evolution

Looped Transformers create a useful train- and test-time compute axis by reusing the same Transformer block over recurrent depth, increasing effective depth at a fixed parameter count. However, that shared block must then govern an entire trajectory of varying hidden states over trained and extrapolated depths. Furthermore, in additive-injection looped Transformers, an input-conditioned signal is reintroduced at every recurrent step, so applying the shared transition at an input-conditioned reference can still move the hidden state. In this paper, we propose Source-Centered State Evolution (SCSE), which is designed to reconcile input conditioning with reference-preserving shared recurrence. Specifically, SCSE retains input dependence through its learned anchor and initial deviation, allows nonzero deviations to drive recurrent computation while mapping zero deviation to zero, and guarantees exact anchor invariance through its zero-deviation mask. The designated anchor is thereby a one-step fixed point by construction. The zero-deviation forcing bias is the next deviation produced from the anchor itself and vanishes in SCSE, while nonzero deviations remain active and support state-dependent recurrent computation. Our theory shows that the zero-deviation forcing bias is a design degree of freedom whose task effect can be harmful, neutral, or beneficial; SCSE resolves this choice in favor of exact anchor invariance by setting the bias to zero. Across WikiText-2, WikiText-103, direct web-corpus pretraining, held-out web-text transfer, and LAMBADA completion, SCSE improves the controlled recurrent quality frontier. Ablation studies identify the learned anchor and the anchor-coordinate deviation recurrence as the primary contributors to the gain, and a trained-model case study grounds the anchor-response diagnostic in observed recurrent motion.

Lottery Tickets Are Not Deployment Tickets

Reports on how sparsification, compression, and lottery tickets change model behavior have been mixed in the prior literature, with beneficial effects observed in some studies and adverse effects in others. Moreover, prior work has not considered actual deployment conditions, where decision logic is already fixed for the incumbent. To assess these mixed findings from a practical standpoint, we study the production-replacement question at the deployment level, namely whether an accuracy-matched lottery ticket or another sparse challenger can replace an incumbent dense model without reconfiguring downstream decision logic. We therefore audit a broad, protocol-specific panel of deployment-relevant behaviors spanning calibration, OOD response, class-level reliability, representations, and downstream policy decisions, and summarize clean-accuracy-excluded deviations with a behavioral-compatibility distance. Across extensive experiments, sparse candidates repeatedly recover dense-reference accuracy yet remain behaviorally different; in several study-band-matched settings, LTs also show lower corruption accuracy. In small-gap settings with fixed-threshold policy diagnostics, lottery-ticket replacement changes 7% to 10% of accept--review decisions. This churn creates precisely the burden that drop-in replacement is meant to avoid: reconfiguring and revalidating downstream decision logic. These findings establish the limits of clean-accuracy certification: Establishing compatibility with a fixed incumbent is distinct from attributing churn uniquely to sparsity or treating every measured deviation as harmful. Our theory explains the routing result: Even exact pointwise top-1 agreement cannot bound fixed-threshold decision changes, and small confidence shifts near the operating boundary can generate first-order routing churn.

SHiPPO: Recurrent Memory with Transported Polynomial Projections

HiPPO gives recurrent states memory semantics as coefficients of online polynomial projections, but in fixed channel coordinates. Modern selective SSMs, by contrast, rely on token-dependent control and channel interaction. We introduce SHiPPO (Sylvester HiPPO), a transported projection-memory prior that lifts HiPPO coefficient memories into a moving channel frame. For any fixed or realized right-transport path, SHiPPO transports the approximation family and channel metric together; conditional on that path, the state is ordinary HiPPO in a tied moving frame and follows Sylvester coefficient dynamics, preserving the left online-memory operator while adding right-action transport. For selective-SSM execution, we derive a restricted group-local realization with controller-compatible right actions, exponential-adjusted updates, exact block-affine scan, and recurrent decoding. We also give a simultaneous-reducibility criterion identifying when right transports collapse to static mixing plus independent scalar or blockwise banks. Controlled diagnostics show that larger current-token write rank improves ordinary prediction error but cannot recover order-sensitive changes to already-written memory; transported-memory variants recover this signal, which disappears when the transport pathway is removed. A finite-field associative-recall diagnostic with interleaved bindings, operations, and queries provides complementary autoregressive evidence while leaving the preferred right-action realization open. Taken together, these results support SHiPPO as a mechanistically grounded transported-memory prior, with evidence focused on memory mechanisms rather than broad sequence-modeling dominance.

Algorithmic Foundations of Deep Learning: Complexity-Theoretic Rates and a Characterization of Universal Approximation

Feedforward neural network (NN) expressivity is typically studied by emulating optimal basis-expansion schemes. While powerful, this perspective is incomplete: it primarily captures complexity through regularity, and therefore does not distinguish intuitively simple and complicated objects with comparable regularity, such as the square-root function and a typical Brownian path. The guiding message is that neural networks should be viewed not only as flexible basis functions, but also as models of computation. If a function is computable by a real-valued circuit over a prescribed elementary gate language, then it can be computed to comparable accuracy by an NN with explicit depth, width, and non-zero-parameter bounds controlled by the depth, width, gate count, and gate structure. Thus, neural-network complexity is not governed by regularity alone, but also by algorithmic complexity. We then show that any definable NN model satisfying a natural parallelization condition, allowing possibly multivariate non-linearities such as attention or layer normalization, is a universal approximator if and only if it contains a non-affine nonlinearity. The scope of our theory is illustrated by deducing universal approximation guarantees for continuous functions, minimax-optimal approximation guarantees for Besov classes, logarithmic-error complexity for holomorphic functions, and by showing that NNs can emulate numerical algorithms such as Newton-Raphson root finding and power iteration without architecture-specific arguments. Its precision is illustrated by shortest-path computation on $k$-vertex graphs: compiling the tropical dynamic-programming circuit yields NNs with O(log(1/ε)) non-zero parameters, exponentially improving in 1/ε over the generic $O(ε^{-c k^2})$ Lipschitz-approximation scale, for a constant c>0.

EqOD: Symmetry-Informed Stability Selection for PDE Identification

Data-driven identification of partial differential equations (PDEs) relies on sparse regression over a candidate library of differential operators, where larger libraries inflate false positives under observation noise and smaller libraries risk missing true terms. We introduce Equivariant Operator Discovery (EqOD), a fully automatic method combining two library reduction mechanisms. When Galilean invariance is detected from trajectory data via a weak-form structural test, EqOD uses the symmetry-reduced library, eliminating terms that our Galilean exclusion result proves to be absent from the governing equation. Otherwise, it applies randomized LASSO stability selection guided by classical false-positive bounds. A residual-based fallback prevents degradation below the full-library baseline. On 8 PDEs at 4 noise levels, EqOD attains $F_1 = 1.000 \pm 0.000$ on Heat at $20\%$ noise, where WF-LASSO obtains $0.475 \pm 0.181$, official PySINDy 2.0 obtains $0.000$, and the WSINDy reimplementation obtains $0.789$. Under the strict criterion that the mean F1 difference exceeds the larger of the two standard deviations, EqOD wins 7 of 32 cells. WF-LASSO wins none, and the remaining 25 cells are ties. Across all 32 cells, EqOD outperforms PySINDy 2.0.0 in 23 of 32 cells, and all 5 PySINDy wins occur on reaction PDEs. External validation on WeakIdent and PINN-SR datasets gives $F_1 = 1.000$ on all 5 clean benchmarks. NLS, 2D, coupled-system, and cylinder-wake extensions are reported. The Galilean library reduction is proved under explicit autonomy and library assumptions. The stability-selection step is motivated by classical false-positive bounds, while formal guarantees for correlated PDE design matrices remain open.

Per-Loss Adapters for Gradient Conflict in Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) train a single neural approximation by minimizing multiple physics- and data-derived losses, but the gradients of these losses often interfere and can stall optimization. Existing remedies typically treat this pathology either through scalar loss balancing or full-parameter-space gradient surgery, leaving it unclear which intervention is most appropriate. We show that PINN gradient conflict is not a uniform failure mode with one universal remedy. Instead, we identify distinct PINN gradient-conflict regimes, each associated with a different intervention class. Persistent directional conflict may require separate loss-indexed parameter subspaces, magnitude imbalance often favors scalar reweighting, and low or transient conflict may require no extra mitigation. To select between scalar reweighting and a lightweight architectural intervention, we propose a diagnostic-first framework. It profiles a 1000-step unmodified PINN run and, when intervention is warranted, uses one low-rank adapter per loss to create explicit loss-indexed parameter subspaces attached to a shared PINN trunk, providing each loss with a direct gradient pathway. Across more than 60 PDE configurations, including forward, inverse, multi-physics, parameter-varying, and high-dimensional problems up to 50D, persistent directional conflict dominates standard forward $K=3$ benchmarks and a natural $K=4$ thermoelastic system, where adapters combined with reweighting yield significant improvements. In contrast, $K=3$ inverse problems and natural $K=5$ and $K=6$ multi-physics systems are largely magnitude-dominated and often favor reweighting alone, while full-parameter-space gradient surgery can fail on heterogeneous parameter spaces.

Residual Koopman Spectral Profiling for Predicting and Preventing Transformer Training Instability

Training divergence in transformers wastes compute, yet practitioners discover instability only after expensive runs begin. They therefore need an expected probability of failure for a transformer before training starts. Our study of Residual Koopman Spectral Profiling (RKSP) provides such an estimate. From a single forward pass at initialization, RKSP extracts Koopman spectral features by applying whitened dynamic mode decomposition to layer-wise residual snapshots. Our central diagnostic, the near-unit spectral mass, quantifies the fraction of modes concentrated near the unit circle, which captures instability risk. For predicting divergence across extensive configurations, this estimator achieves an AUROC of 0.995, outperforming the best gradient baseline. We further make this diagnostic actionable through Koopman Spectral Shaping (KSS), which reshapes spectra during training. We empirically validate that our method works in practice: RKSP predicts divergence at initialization, and when RKSP flags high risk, turning on KSS successfully prevents divergence. In the challenging high learning rate regime without normalization layers, KSS reduces the divergence rate from 66.7% to 12.5% and enables learning rates that are 50% to 150% higher. These findings generalize to WikiText-103 language modeling, vision transformers on CIFAR-10, and pretrained language models, including GPT-2 and LLaMA-2 up to 7B, as well as emerging architectures such as MoE, Mamba-style SSMs, and KAN.

Radial Müntz-Szász Networks: Neural Architectures with Learnable Power Bases for Multidimensional Singularities

Radial singular fields, such as $1/r$, $\log r$, and crack-tip profiles, are difficult to model for coordinate-separable neural architectures. We show that any $C^2$ function that is both radial and additively separable must be quadratic, establishing a fundamental obstruction for coordinate-wise power-law models. Motivated by this result, we introduce Radial Müntz-Szász Networks (RMN), which represent fields as linear combinations of learnable radial powers $r^μ$, including negative exponents, together with a limit-stable log-primitive for exact $\log r$ behavior. RMN admits closed-form spatial gradients and Laplacians, enabling physics-informed learning on punctured domains. Across ten 2D and 3D benchmarks, RMN achieves 1.5$\times$--51$\times$ lower RMSE than MLPs and 10$\times$--100$\times$ lower RMSE than SIREN while using 27 parameters, compared with 33,537 for MLPs and 8,577 for SIREN. We extend RMN to angular dependence (RMN-Angular) and to multiple sources with learnable centers (RMN-MC); when optimization converges, source-center recovery errors fall below $10^{-4}$. We also report controlled failures on smooth, strongly non-radial targets to delineate RMN's operating regime.