✗ Failed on benchmark
2026
Use the paper's explicit compact factor of a symplectic state-transition matrix to measure aggregate rotation speed in hidden-state dynamics. Penalize excessive or rapidly varying angular velocity rather than penalizing the full recurrent matrix, preserving nontrivial Hamiltonian rotations while suppressing phase drift that can destabilize long sequences.
Useful7/10
Difficulty5/10
Novelty7/10
✗ Failed on benchmark
2026
Implement a momentum optimizer as a contact Hamiltonian splitting rather than as a direct Euler discretization. Introduce an auxiliary scalar contact state and compose exact kinetic, potential, and damping subflows; this produces a second-order conformal integrator whose modified contact energy should decay more reliably at moderately large learning rates.
Useful7/10
Difficulty5/10
Novelty7/10
△ Mechanism confirmed, baseline not beaten
2026
Replace an unconstrained recurrent transition or latent ODE vector field with a port-Hamiltonian update whose metric is positive definite and whose interaction operator is skew-symmetric. Use an implicit midpoint step so the quadratic latent energy is preserved exactly in the unforced, constant-metric case, preventing long-horizon drift while retaining learnable nonlinear interactions.
Useful7/10
Difficulty5/10
Novelty5/10
✓✓ Beats tuned baseline
2026
Replace an unconstrained one-step transition network with a symmetric damping–symplectic-core–damping composition. The damping strength is one learned scalar rate and is applied through positive exponential diagonal factors, so every step has a known contraction law while the neural core models nonlinear conservative transport.
Useful7/10
Difficulty5/10
Novelty7/10
Unverified
2026
Normalize a learned symplectic recurrent transition using the paper's explicit positive-definite factor instead of projecting the entire matrix onto an orthogonal group. Retain the compact unitary dynamics as the stable transport component and separately damp only the noncompact factor, providing a tunable stability mechanism that preserves symplectic structure.
Useful6/10
Difficulty6/10
Novelty5/10
Unverified
2026
Replace an unconstrained recurrent or residual linear transition with a matrix generated through the paper's twisted Cayley chart and exact exponential flow. The layer evolves a constrained operator analytically rather than learning arbitrary weights, while retaining trainable symmetric chart coordinates and a continuous time-scale parameter.
Useful6/10
Difficulty6/10
Novelty6/10
Unverified
2026
Represent each token value in an attention head as a bounded three-dimensional Lie-algebra vector and impose a weighted polygon-closure condition on the aggregate value vectors. The module is invariant to a common \(SU(2)\simeq SO(3)\) rotation of all token vectors, preventing the head from spending capacity on an arbitrary global orientation. A soft closure penalty gives a drop-in experiment, while projection onto the zero-sum manifold provides a harder constrained variant.
Useful6/10
Difficulty5/10
Novelty7/10
Unverified
2026
Replace a conventional momentum update by a damped second-order trajectory with a configuration-dependent dense kinetic metric. Evolve two phase-space copies using symmetric split orderings, project both copies exactly back to their averaged physical state, and apply exact friction half-steps so momentum decay remains stable at large step sizes.
Useful6/10
Difficulty6/10
Novelty6/10
Unverified
2026
Compose independently parameterized neural dynamical modules through power-preserving skew coupling instead of equality penalties or projected constraints. This creates a modular graph or world model in which information exchanged between modules is antisymmetric, so internal coupling cannot create or destroy total latent energy.
Useful6/10
Difficulty6/10
Novelty6/10
Unverified
2026
Replace explicit quotient construction by a differentiable projection that removes learned group-orbit directions from both source and target features. The paper's reduction argument shows that a closed level constraint makes the restricted form horizontal, so the network can operate on invariant coordinates while retaining a measurable residual for symmetry leakage.
Useful6/10
Difficulty6/10
Novelty5/10
Unverified
2026
Construct a hybrid neural ODE from several smooth vector-field branches and select the active branch using a learned Hamiltonian-like score. Track a positive-definite matrix representing local tangent sensitivity and force its discrete evolution to be positive semidefinite, adapting the paper's monotone Jacobi-curve condition to neural dynamics.
Useful6/10
Difficulty6/10
Novelty7/10
△ Mechanism confirmed, baseline not beaten
2026
Use a symplectic Hamiltonian update as a recurrent or state-space neural block, preserving a learned modified energy across many layers or time steps. This targets residual and recurrent architectures where ordinary Euler updates accumulate drift during long rollouts.
Useful6/10
Difficulty5/10
Novelty5/10
△ Mechanism confirmed, baseline not beaten
2026
Augment neural-network parameters with momentum variables and update the pair using a symplectic map generated by a Hamiltonian. The optimizer approximately preserves a modified Hamiltonian, reducing systematic energy drift and potentially making long unrolled optimization more stable.
Useful6/10
Difficulty4/10
Novelty4/10
△ Mechanism confirmed, baseline not beaten
2026
Augment a latent neural ODE with learned constraint functions whose time derivatives are forced to close linearly on the constraint family, making the zero level set invariant by construction. Integrate only the quotient-relevant coordinates while treating the constraint-generated characteristic coordinates as gauge variables, reducing latent dimension and suppressing long-horizon constraint drift.
Useful6/10
Difficulty6/10
Novelty6/10
✗ Mechanism failed
2026
Replace an unconstrained latent transition by a layer with a distinguished scalar coordinate \(t\) and a symplectic leaf state \(x=(q,p)\). The layer advances \(t\) through a Reeb drift while updating \(x\) with a symplectic Hamiltonian step, preventing arbitrary mixing between progression and content coordinates and potentially improving long-horizon stability.
Useful6/10
Difficulty6/10
Novelty6/10
✗ Mechanism failed
2026
Replace standard heavy-ball momentum with an update derived from a discrete kinetic-minus-loss action and a discrete viscous force. The force discretization produces a rational damping factor that remains controlled over a specified range of step sizes, potentially reducing oscillations and instability without Adam-style second-moment state.
Useful6/10
Difficulty4/10
Novelty6/10
Unverified
2026
Represent a time-dependent Hamiltonian system on the reduced state $(q,t,p_q)$ rather than on the redundant extended state $(q,t,p_q,p_t)$. A neural Hamiltonian section predicts one canonical representative of each affine cotangent fiber, while an optional symmetry loss enforces consistency under transformations that translate time.
Useful5/10
Difficulty5/10
Novelty8/10
Unverified
2026
Replace an unconstrained residual block by a four-field feature dynamics containing a primary feature T, flux-like auxiliary features J, curl-cleaning features psi, and a scalar cleaning feature phi. Couple these fields with learned skew-adjoint spatial operators so that the reversible block preserves the squared feature norm, while a separately controlled relaxation term can remove high-frequency or constraint-violating components. Use an exact Cayley update rather than explicit Euler to…
Useful5/10
Difficulty6/10
Novelty5/10
Unverified
2026
Construct the latent transition from a nondegenerate bilinear form phi and a form-compatible operator instead of from an unconstrained dense matrix. The resulting SSM has an exact orthogonal or symplectic algebraic structure, reducing transition parameter redundancy and testing whether preservation of a latent pairing improves extrapolation on reversible, parity-sensitive, or Hamiltonian-like sequence tasks.
Useful5/10
Difficulty5/10
Novelty5/10
Unverified
2026
Replace an unconstrained recurrent transition on a state (q,p) with a discrete variational transition generated by a strictly convex distance-like function L(q,q_1). The next state is found from the implicit reflection equation L_2(q,q_1)+L_1(q_1,q_2)=0, while the induced two-form is preserved by construction; this should reduce energy-like drift and exploding or vanishing sensitivity over long sequences.
Useful5/10
Difficulty6/10
Novelty4/10
Unverified
2026
Replace an ordinary input-convex potential with a potential whose Hessian is encouraged to be symmetric positive definite and symplectic. Add a curvature penalty based on the scalar curvature of the Hessian metric, together with a theorem-derived interior target proportional to the inverse squared distance to the domain boundary. This should suppress pathological third-derivative oscillations while preserving nonquadratic structure near boundaries.
Useful5/10
Difficulty7/10
Novelty8/10