SymFlux: ML Framework & Symplectic Invariant

• SymFlux is a dual framework that integrates a deep learning model for Hamiltonian recovery with an abstract symplectic invariant derived from Fukaya categories.
• The deep learning approach employs a hybrid CNN–LSTM architecture to accurately reconstruct closed-form Hamiltonians from vector field images, achieving up to 88% exact match on benchmarks.
• The SymFlux group invariant, based on Hochschild cohomology, reveals hidden symmetries and distinguishes subtle symplectic phenomena where classical flux methods fall short.

SymFlux refers to two conceptually distinct but thematically related constructs in symplectic geometry and Hamiltonian dynamics: (1) the deep learning framework SymFlux for symbolic regression of Hamiltonian functions from vector fields, and (2) the abstract symplectic invariant termed the “SymFlux group” or “SymFlux invariant,” derived from Hochschild cohomology classes in the context of Fukaya categories. Both lines of development target the interface between symplectic structure, invariants, and computational or categorical methodology, with a focus on uncovering hidden symmetries, conserved quantities, or homological obstructions.

1. Foundations in Symplectic Geometry and Hamiltonian Dynamics

A symplectic manifold $(M, \omega)$ is a $2n$-dimensional smooth manifold equipped with a closed, non-degenerate $2$-form $\omega$. On the standard symplectic plane $M = \mathbb{R}^2$ with coordinates $(q, p)$, the canonical form is $\omega_0 = dq \wedge dp$. Given a smooth function $H: \mathbb{R}^2 \rightarrow \mathbb{R}$ (the Hamiltonian), the corresponding vector field $X_H$ satisfies $\iota_{X_H}\omega_0 = dH$, or, equivalently, $2n$0 with $2n$1 and $2n$2. Hamilton's equations are given by: $2n$3 The inverse problem—Hamiltonization—asks: given an (approximate) vector field $2n$4, can one find $2n$5 such that $2n$6? This Hamiltonization problem is fundamental in mechanics and geometry for uncovering conserved quantities such as total energy and is central to the development of symplectic invariants.

The classical flux homomorphism, for a loop $2n$7 in the identity component of the symplectomorphism group $2n$8, is defined via $2n$9 by $2$0 and $2$1, leading to the cohomology class

$2$2

However, this detects only degree-$2$3 classes and is trivial when $2$4. This motivates seeking more refined invariants and computational tools.

2. The SymFlux Deep Learning Framework: Automated Hamiltonization

The SymFlux framework (Evangelista-Alvarado et al., 8 Jul 2025) represents a computational approach to symbolic Hamiltonization via deep learning and symbolic regression. The algorithmic architecture consists of a hybrid CNN–LSTM system, trained to reconstruct closed-form Hamiltonians from visual representations of their vector fields.

Input Encoding: Each sample is encoded as a multi-channel image $2$5 comprising:

• Quiver plots (arrows depicting $2$6 direction),
• Streamlines (integrated trajectories for qualitative structure),
• Heatmaps of $2$7 (magnitude).

Feature Extraction: Images are processed using one of three variants:

• SymFlux-CNN (custom small CNN),
• SymFluxRN (ResNet backbone),
• SymFluxX (Xception backbone).

The CNN produces a feature vector $2$8.

Tokenization and LSTM Decoding: Hamiltonians $2$9 are represented as sequences of tokens from a basis $\omega$0 of monomials and trigonometric functions. An LSTM decoder, conditioned on $\omega$1, outputs the sequence representing $\omega$2 token by token.

Loss and Optimization: Training employs categorical cross-entropy loss at each sequence step; optimization uses Adam with learning rate $\omega$3 and selection via Keras Tuner.

A key aspect is the construction of synthetic datasets $\omega$4, where Hamiltonians from all nontrivial linear combinations of the basis are symbolically differentiated to obtain $\omega$5, evaluated on sample grids, and visualized as images. The visual datasets (train/test split: 75/25) enable supervised training purely from synthetic data.

3. Evaluation Metrics and Empirical Performance

Conventional string metrics were found inadequate for comparing symbolic outputs. SymFlux introduces the Euclidean token distance in one-hot encoding space: $\omega$6 where $\omega$7 represent ground-truth and predicted tokens.

Empirical results on synthetic benchmarks (e.g., $\omega$8) yield:

• SymFluxRN (ResNet): 88% exact match,
• SymFluxX (Xception): 88%,
• SymFlux (custom CNN): 87%.

Illustrative examples:

• The harmonic oscillator $\omega$9 is recovered exactly.
• The pendulum $M = \mathbb{R}^2$0 is predicted as $M = \mathbb{R}^2$1, resulting in nearly identical vector fields.
• Non-polynomial Hamiltonians (e.g., Lotka–Volterra, SIS epidemiological models) are approximated by the nearest valid form within the basis.

The frameworks demonstrate that, when the true Hamiltonian lies within the span of the learned vocabulary, recovery is highly accurate.

4. The Abstract SymFlux Group as a Symplectic Invariant

Independently, the “SymFlux group” (Seidel, 2011) arises as an invariant in symplectic topology, constructed from the Hochschild cohomology of the (curved) Fukaya $M = \mathbb{R}^2$2–category $M = \mathbb{R}^2$3 of a symplectic manifold $M = \mathbb{R}^2$4. This construction generalizes classical flux:

• Quantum cohomology $M = \mathbb{R}^2$5 with Novikov field $M = \mathbb{R}^2$6 and quantum product “$M = \mathbb{R}^2$7”.
• Curved Fukaya category parametrized by $M = \mathbb{R}^2$8 and $M = \mathbb{R}^2$9.

The open–closed string map $(q, p)$0 sends odd-degree classes to deformation fields.

A class $(q, p)$1 is called periodic if for all perfect modules over the associated $(q, p)$2–category, suitable families exist with trivialized induced connections. The set of such periodic classes is denoted $(q, p)$3, where $(q, p)$4 parameterizes family structures (e.g., elliptic curves with holomorphic forms).

Main Theorems:

• The SymFlux group $(q, p)$5 is invariant under symplectic isotopy and quasi-equivalences.
• Functoriality holds for right-perfect bimodules and push-forward along symplectic correspondences, preserving SymFlux invariants under symplectic embeddings and blow-ups.

5. Exemplar Applications and Distinguishing Power

SymFlux Deep Learning (Automated Discovery)

Applications extend beyond mechanics to fluid models, plasma, and dynamical systems where the underlying vector field is empirically observable but the Hamiltonian structure is uncertain. The framework provides interpretable, closed-form expressions, thus facilitating physics-informed modeling.

SymFlux Group (Symplectic Geometry)

• Symplectic Mapping Tori: The periodicity of classes such as $(q, p)$6 distinguishes the trivial torus bundle from mapping tori with nontrivial dynamics (e.g., those resulting from Dehn twists), independent of classical flux, even in the $(q, p)$7 regime.
• Blow-ups: The abstract SymFlux invariant detects differences between symplectic structures arising from blow-ups, distinguishing mutation-equivalent but non-isomorphic forms. For example, the periodicity of $(q, p)$8 is preserved in trivial cases but lost for nontrivial blow-ups under specified Gromov–Witten assumptions.

A summary of core constructs and distinctions:

Construct	Domain	Detects
SymFlux (CNN–LSTM SR)	Computational/ML	Hamiltonian function from vector field images
SymFlux group $(q, p)$9	Symplectic topology, Fukaya cat	Periodic classes, homological invariants
Classical flux homomorphism	Symplectic topology	$\omega_0 = dq \wedge dp$0 classes from loops

6. Limitations, Extensions, and Open Questions

SymFlux (Deep Learning)

• Recovery is limited by the basis: functions outside of the selected vocabulary (logarithms, inverses, special functions) cannot be reconstructed exactly.
• Training data are synthetic; real physical vector fields may require denoising, more flexible sampling, and potentially richer architectures (e.g., Transformer decoders).
• Adding visual channels (e.g., divergence maps, phase portraits) or expanding the token set are potential avenues for greater generality.

SymFlux Group

• When $\omega_0 = dq \wedge dp$1 admits circle actions or automorphism families with vanishing classical flux, the degree-$\omega_0 = dq \wedge dp$2 SymFlux group recovers classical periodicities studied by Fuks–McDuff–Schwarz.
• In cases where $\omega_0 = dq \wedge dp$3 and classical flux vanishes, higher odd-degree classes $\omega_0 = dq \wedge dp$4 detected via open–closed string maps provide new invariants.
• Conjectured extensions include computation entirely in quantum cohomology under homological mirror symmetry and development of non-archimedean analytic versions (families over annuli) that could reveal the convergence properties of Floer-theoretic invariants.
• Understanding mutation and structural change under surgeries (Lagrangian nodal sliding, coisotropic blow-downs) remains open.

A plausible implication is that the two “SymFlux” paradigms, though independently realized—one as a ML-based symbolic regression tool, the other as a categorical invariant—both signify the emerging interplay between computational, categorical, and geometric viewpoints in modern symplectic theory (Seidel, 2011, Evangelista-Alvarado et al., 8 Jul 2025).