cs.AIMar 31, 2026

Metriplector: From Field Theory to Neural Architecture

Authors: Dan Oprisa, Peter Toth

Organizations: Spheroid Labs∗

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Abstract

We present Metriplector, a neural architecture primitive in which the input configures an abstract physical system -- fields, sources, and operators -- and the dynamics of that system is the computation. Multiple fields evolve via coupled metriplectic dynamics, and the stress-energy tensor T^{μν}, derived from Noether's theorem, provides the readout. The metriplectic formulation admits a natural spectrum of instantiations: the dissipative branch alone yields a screened Poisson equation solved exactly via conjugate gradient; activating the full structure -- including the antisymmetric Poisson bracket -- gives field dynamics for image recognition, language modeling, and robotic control. We evaluate Metriplector across five domains, each using a task-specific architecture built from this shared primitive with progressively richer physics: 81.03% on CIFAR-100 with 2.26M parameters; 88% CEM success on Reacher robotic control with under 1M parameters; 97.2% exact Sudoku solve rate with zero structural injection; 1.182 bits/byte on language modeling with 3.6x fewer training tokens than a GPT baseline; and F1=1.0 on maze pathfinding, generalizing from 15x15 training grids to unseen 39x39 grids.

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Figure 1: Metriplector field interaction. Top: KK fields ψk\bm{\psi}_{k} evolve via metriplectic dynamics over the spatial grid (gradient arrows show ∇ψk\nabla\bm{\psi}_{k} ); the outer product ∇ψa⊗∇ψb\nabla\bm{\psi}_{a}\otimes\nabla\bm{\psi}_{b} yields three stress-energy components. Bottom: per-field gradient energy Ek=∣∇ψk∣2E_{k}=|\nabla\bm{\psi}_{k}|^{2} , cross-field correlation Eab=∇ψa⋅∇ψbE_{ab}=\nabla\bm{\psi}_{a}\cdot\nabla\bm{\psi}_{b} , and vorticity Vab=∇ψa×∇ψbV_{ab}=\nabla\bm{\psi}_{a}\times\nabla\bm{\psi}_{b} , summed and projected via Conv 1×11{\times}1 into h\mathbf{h} . Shown for K=3K{=}3 ; the full model uses K=32K{=}32 .
Figure 2: The metriplectic spectrum. All five domains instantiate the same GENERIC equation. Maze and Sudoku use only the dissipative branch ( MM ), solved at equilibrium via CG. Language uses causal dissipation via scan. CIFAR-100 and Reacher activate full metriplectic structure via Euler integration; Reacher additionally incorporates canonical symplectic JΩJ_{\Omega} , port-Hamiltonian action conditioning, and multi-step AR training.
 AttentionMetriplector
DynamicsPure dissipation ( ∇H\nabla H only)Designed metriplectic ( M+JM{+}J )
HamiltonianLogSumExp (nonlinear)Spatial (quadratic) + stress-energy
Spatial couplingGlobal (all-to-all in standard form)Local, O(9N)O(9N) DWConv 3×33{\times}3
Channel mixingMLP ( O(D2)O(D^{2}) )Complete JJ + stress-energy ( O(K2)O(K^{2}) )
ConservationNoneEnergy conservation by antisymmetric JJ
Cross-field coverage—All K(K−1)/2K(K{-}1)/2 pairs
Table 1: Attention vs. Metriplector as dynamical systems. Each layer’s update Δψ\Delta\bm{\psi} can be decomposed into dynamics type (how it moves on the energy landscape) and Hamiltonian type (what the landscape looks like). DD = representation dimension, KK = field dimension, NN = spatial positions.
Figure 3: Metriplector architecture. A single round of the recurrent V-cycle (repeated RR times with shared weights). The cell encoder produces per-cell features h\mathbf{h} from input, previous predictions, position, and round fraction. Learned symmetric conductances wijw_{ij} define the graph Laplacian. Damping and source MLPs produce per-cell screening and forcing terms. KK independent screened Poisson equations are solved via CG. The dissipation readout Dk=∑jwij(ψi−ψj)2≈∣∇ψk∣2D_{k}=\sum_{j}w_{ij}(\bm{\psi}_{i}-\bm{\psi}_{j})^{2}\approx|\nabla\bm{\psi}_{k}|^{2} extracts the diagonal of the stress-energy tensor (the same TμνT^{\mu\nu} readout used in CIFAR-100, cf. Section 5 ); 8-way directional scans add global context. The multigrid object layer restricts to learned soft-assignment groups, solves a coarse Poisson, and prolongates back. The decoder reads all features to produce per-cell predictions, fed back via annealed softmax ( τ ⁣:1.0→0.2\tau\!:1.0\to 0.2 ). Blue : prediction feedback across rounds. Orange : multigrid V-cycle within each round. Reasoning domains (maze, Sudoku) share this CG-based architecture with different KK , RR , and output dimension CC ; CIFAR-100 uses a distinct Euler-based instantiation (Section 3.4 ); Reacher extends the CIFAR instantiation with port-Hamiltonian action conditioning for world modeling (Section 3.5 ).
DomainGraphKKDepthKobjK_{\text{obj}}OutputParamsDynamics
CIFAR-1002D grid 16 ×\times 1632L=12L{=}12—100 classes2.26MEuler, M+LM{+}L
Reacher2D grid 8 ×\times 816L=4L{=}4—8 ×\times 8 ×\times 128 latent913KEuler, JΩ+MJ_{\Omega}{+}M , port-Ham, AR
Sudoku8-conn lattice16R=32R{=}32169 digits120KCG, MM only
LanguageCausal 1D chain16L=6L{=}6—1024 tokens18.5MScan, MM causal
Maze4-conn grid2R=1R{=}1—5 classes43.8KCG, MM only
Table 2: Domain configurations. All variants share the metriplectic formalism; domains differ in which GENERIC channels are active, the solver, and domain-specific adaptations. Sudoku and CIFAR-100 implement the same underlying physics (Section 5 ): both read TμνT^{\mu\nu} from evolved fields. Reacher extends this to action-conditioned dynamics.
Figure 4: CIFAR-100 metriplectic layer ( ×12\times 12 , non-shared weights). The representation h\mathbf{h} ( D=128D{=}128 ) flows along the top via residual connections. Each layer projects h\mathbf{h} down to K=32K{=}32 physics fields ψ\bm{\psi} , evolves them under the full metriplectic equation (diffusion ++ advection ++ damping ++ source), extracts physically meaningful features via the stress-energy tensor (Noether readout), and projects back to DD via gated mixing. The KK -field bottleneck reduces pairwise interaction cost by 16×16\times compared to operating in the full DD space.
Figure 5: Reacher world model architecture. A convolutional encoder maps 64×6464{\times}64 images to an 8×8×1288{\times}8{\times}128 latent grid. The action modulates PDE operators via port-Hamiltonian conditioning (zero-initialized). Four MetriplectorLayers evolve fresh ψ\bm{\psi} fields—organized as 8 conjugate (q,p)(q,p) pairs with canonical symplectic JΩJ_{\Omega} —and extract TμνT^{\mu\nu} features. Multi-step AR training (scheduled warmup) closes the exposure bias gap. The prediction target comes from an EMA target encoder. At evaluation, CEM plans by rolling out action sequences through the latent dynamics.
Figure 6: Causal Poisson language model. Tokens are embedded and passed through L=6L{=}6 non-shared CausalPoissonLayers. Each layer solves the causal Poisson recurrence via O(Nlog⁡N)O(N\log N) parallel associative scan, applies progressive multigrid (token →\to chunk →\to section scales with shifted pooling for causal safety), computes cross-field outer products ( ψ⊗ψ\bm{\psi}\otimes\bm{\psi} ), and integrates all features into the hidden state h\mathbf{h} via a round MLP with residual connection.
ModelParamsAccNote
Metriplector (ours, full M+LM{+}L )
Metriplectic K=32K{=}32 s=8s{=}82.26M81.03TμνT^{\mu\nu} readout, 8 sub
Metriplectic K=32K{=}32 s=3s{=}32.26M80.63 substeps
Metriplectic K=32K{=}32 s=1s{=}12.26M80.41 substep
Noether K=32K{=}321.70M79.8Position readout
PhysicsSelects-V2 K=32K{=}321.49M78.4Curvature readout
Table 3: CIFAR-100 results. All Metriplector models: 12 layers, patch size 2, identical recipe (Adam ( Kingma & Ba, 2015 ) , lr= 3×10−33{\times}10^{-3} , cosine, AutoAugment ( Cubuk et al., 2019 ) , label smoothing 0.1, 200 epochs).
Figure 7: CIFAR-100: Accuracy vs. parameters. Metriplector variants (blue) achieve 80%+ accuracy with 2.26M parameters—10–15 ×\times fewer than conventional architectures (gray) at similar accuracy levels.
AblationAcc (%)Δ\Delta from base
Cumulative improvements (from V1 = 77.6%)
+ SiLU + BN residual engineering (V1 →\to V2)78.4+0.8
+ 3 Euler substeps (more propagation per layer)78.6+1.0
+ Noether currents ( px,py,L,Dp_{x},p_{y},L,D , position-aware)79.8+2.2
+ Stress-energy + 8 substeps ( ∇ψ×∇ψ\nabla\bm{\psi}{\times}\nabla\bm{\psi} , K2K^{2} features)81.03+3.4
Component removal (from V2 baseline = 78.4%)
Table 4: CIFAR-100 ablations. Top : cumulative improvements from V1 (77.6%); each row adds one component on top of all previous rows; Δ\Delta is relative to V1. Bottom : component removal from V2 baseline (78.4%); Δ\Delta is the accuracy drop when that component is removed.
Figure 8: CIFAR-100 ablation impacts. Accuracy improvement from each architectural component relative to the PhysicsSelects-V2 K=32K{=}32 baseline (78.4%). The operator-from- h\mathbf{h} principle (+14.3%) and the Poisson bracket JJ (+13.4%) are the most important factors. Fresh ψ\bm{\psi} per layer (+3.4%) validates the non-recurrent physics design.
Figure 9: CIFAR-100 physics diagnostics across 12 layers. (a) Singular value spectrum of the learned Poisson tensor JJ —values appear in degenerate pairs, the hallmark of skew-symmetric structure. (b) JJ effective rank and null-space dimensions; ∥J∥F\|J\|_{F} grows monotonically with depth, indicating stronger cross-field coupling in later layers. (c) ψ\bm{\psi} magnitude and standard deviation remain stable (0.25–0.5) across L0–L10, with a sharp spike at L11 before the stress-energy readout. (d) Spatial variance increases at the final layer, reflecting transition from distributed texture to localized semantic features.
Figure 10: ψ\bm{\psi} field specialization across 12 layers (32 fields ×\times 12 layers). (a) Field magnitude: most fields remain moderate through L0–L10, with a sharp spike at L11 where specific fields (e.g., k=3,20,21k{=}3,20,21 ) concentrate discriminative features. (b) Spatial variance: early layers develop spatially uniform fields (low variance); L11 shows dramatic spatial structure as the model localizes class-specific features before readout.
Figure 11: CIFAR-100 forward dynamics budget. (a) Hidden state h\mathbf{h} : magnitude and per-layer update Δh\Delta\mathbf{h} both grow monotonically; the step size γ\gamma increases from 0.12 to 0.22, self-organizing an expanding dynamics budget. (b) ψ\bm{\psi} fields: magnitude and standard deviation are stable (0.25–0.5) across L0–L10, with spatial variance spiking at L11.
ModelParamsCEMArchitecture
LeWM∼\sim 15M86%ViT-Tiny + 6-layer transformer
DINO-WM>> 21M79%Frozen DINOv2 ViT-S
PLDM∼\sim 15M78%ViT + GRU predictor
Random—10%—
Ours913K88%4 ×\times MetriplectorLayer
Table 5: Reacher: comparison with baselines. All methods use the qpos_match criterion from Maes et al. (2026) . CEM: 50 episodes, 300 candidates, 30 elite, 30 iterations, horizon 5.
Figure 12: Emergent symplectic layer specialization. (a) Evolution of JspecJ_{\mathrm{spec}} (spectral radius of the full Poisson bracket J=JΩ+JcrossJ=J_{\Omega}+J_{\mathrm{cross}} ) during training. All layers begin near canonical ( Jspec=1.0J_{\mathrm{spec}}=1.0 ) and develop a depth-ordered hierarchy: L2 acquires the richest inter-pair coupling ( ++ 56% beyond canonical). The “AR on” marker at epoch 10 shows the onset of multi-step autoregressive training. (b) Final layer configuration: advection strength α\alpha vs. JspecJ_{\mathrm{spec}} reveals two emergent physical regimes—“fast canonical transport” (L0–L1, high α\alpha ) and “slow precision refinement” (L2–L3, low α\alpha , high coupling)—resembling symplectic splitting integrators ( Yoshida, 1990 ) . Bubble size ∝\propto energy flux ∣dH∣|dH| .
Figure 13: Emergent box discovery in Sudoku. The object layer discovers all 9 Sudoku 3 ×\times 3 boxes from an 8-connected spatial lattice with no box-level supervision. Assignments are sharp ( τ=0.136\tau{=}0.136 , entropy ≈0.032{\approx}0.032 ) and static across all puzzles.
Figure 14: Learned conductance matrix WsymW_{\text{sym}} (Sudoku). Entries show coupling strength between cell types (digits 1–9 and padding). The model learns sparse, structured couplings: strong links (4 ↔\leftrightarrow pad, 2 ↔\leftrightarrow 4, 5 ↔\leftrightarrow 7) emerge without supervision.
ModelParamsBPB ↓\downarrowTokensNote
Metriplector (Stacked v6)18.5M1.1822.0B1 ×\times GH200, 2.6 hr
GPT baseline9.8M1.2247.2B8 ×\times H100, 10 min
PG leaderboard SOTA—1.119—+TTT, EMA, etc.
Table 6: Language modeling results. BPB on FineWeb validation. All models ≤\leq 16 MB.
Figure 15: Language modeling BPB progression. GPT baseline (1.224 BPB, dashed) shown for reference; Metriplector reaches 1.182 BPB with 3.6 ×\times fewer training tokens.
ModelParamsF1 (15 ×\times 15)F1 (39 ×\times 39)Steps
Harmonic baseline00.47——
MetriNet Phase 7.3B9,1000.59—4,000
Metriplector 7.53K ( K=2K{=}2 )43.8K1.000.4310,000
Metriplector 7.53O ( λ/N\lambda/N )43.8K1.000.95–0.9910,000
Table 7: Maze results. F1 score on tree-maze path prediction. Metriplector achieves F1 = 1.0 on 15 ×\times 15 and F1 = 0.95–0.99 on 39 ×\times 39 across seeds.
DomainParamsResultKey Finding
CIFAR-1002.26M81.03% acc10–15 ×\times fewer params vs. DenseNet
Reacher913K88% CEM16 ×\times fewer params than baselines
Sudoku 9 ×\times 9120K97.2% exact9/9 box discovery, zero injection
Language18.5M1.182 BPB3.6 ×\times fewer tokens vs. GPT
Maze 15 →\to 3943.8KF1 = 0.95–0.99Robust size transfer
Table 8: Cross-domain results. One metriplectic formalism, five instantiations.
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