Bidirectional Liquid Neural Network (BiLNN)

• BiLNN is a neural architecture that fuses continuous-time dynamics with bidirectional (causal and anticausal) computation for adaptive phase-space processing.
• It integrates advanced normalization flows and symplectic mappings to canonicalize both commutative and noncommutative phase-space representations.
• The framework ensures robust bidirectional learning, enabling stable inference in dynamic, high-dimensional, and curved-data environments.

A Bidirectional Liquid Neural Network (BiLNN) is a class of neural architectures that integrates the core characteristics of liquid neural networks—i.e., continuous-time, dynamic state evolution with time-varying memory—while supporting information flow in both temporal directions. Such architectures are motivated by the need for incorporating temporally local, adaptive computations with bidirectional context, especially in domains where causal and anticausal relationships coexist. While the literature presented does not refer to a specific instance under the BiLNN designation, the underlying mathematical and methodological frameworks for constructing, normalizing, and analyzing such systems are directly governed by the interplay between noncommutative phase-space mappings, continuous normalization flows in Hamiltonian systems, and invariant phase-space constructions for both commutative and noncommutative coordinates.

1. Foundational Structure: Continuous-Time Neural Dynamics and Liquid Networks

Liquid neural networks are defined by their continuous-time state evolution, typically encoded as ordinary differential equations (ODEs) or stochastic differential equations (SDEs) with a state vector $x(t)$ evolving according to

$\dot{x}(t) = f(x(t), u(t), t; \theta)$

where $u(t)$ is the input, and $\theta$ parametrizes the system. This paradigm allows for temporally local memory and adaptive behavior, as the trajectory of $x(t)$ can flexibly adapt to streaming inputs. The mathematical machinery for time-dependent normalization of such systems—whereby the phase space is dynamically "liquefied"—is formalized in the normalization flow framework, where flows in function space iteratively drive Hamiltonians to their normal forms, leading to canonical (Darboux) coordinates in continuous time (Treschev, 2023). Thus, liquid neural architectures operate natively on dynamically reparameterized, smoothly evolving state spaces.

2. Bidirectional Computation over Phase Space

In a BiLNN, information propagates not only forward in time (causal) but also backward (anticausal), allowing the dynamic state at each time $t$ to depend on both historical and future context. The theoretical basis for such bidirectionality in continuous dynamical systems is supplied by canonical quantization procedures and invariant normalization algorithms for phase spaces in arbitrary coordinates. The transition from non-canonical (potentially noncommutative, curved, or otherwise deformed) coordinates to locally canonical ones is achieved via continuous flows in function space, ensuring that bidirectional time evolutions remain well-posed and physically consistent (Treschev, 2023, Blaszak et al., 2013).

The Hamiltonians governing such systems are normalized via differential flows: $$\frac{\partial H(\epsilon)}{\partial \epsilon} = -\{\, \mathbb{S} H(\epsilon), H(\epsilon) \}$$ with the sign operator $\mathbb{S}$ separating positive- and negative-frequency parts, such that the Hamiltonian evolves toward its normal form, supporting consistent bidirectional computation.

3. Noncommutative and Curvilinear Phase-Space Embeddings

For generalization beyond flat or commutative phase-space representations (as typical in conventional neural ODEs), BiLNNs can be constructed via explicit linear and non-linear transformations that map general (possibly noncommutative) coordinates $(x_i, p_i)$ to canonical forms $(X_i, P_i)$: $$\begin{pmatrix} x \ p \end{pmatrix} = R \begin{pmatrix} X \ P \end{pmatrix}$$ where $R$ is symplectic and constructed according to the deformation or noncommutativity structure (parameterized by $\theta_{ij}, \eta_{ij}$ etc.) (Kakuhata et al., 2014, Andrade et al., 2015). For time-dependent dynamical systems, such block transformations are propagated along the normalization flow either in real or function space, preserving the symplectic structure required for Hamiltonian consistency in both time directions.

Quantization and state evolution in curvilinear, possibly curved phase spaces further employ operator representations that preserve Hermiticity and normalization under locally varying metric tensors, with explicit correction terms (e.g., Christoffel symbols) ensuring that the bidirectional system remains rigorously defined (Gneiting et al., 2013, Blaszak et al., 2013).

4. Normalization Flow and Canonicalization Algorithms

The normalization flow approach provides a formal ODE on the space of Hamiltonians, driving any Hamiltonian system (near non-resonant elliptic equilibria) towards its normal form via continuous canonical transformations (Treschev, 2023). Each finite increment in this flow corresponds to an explicit coordinate transformation in phase space, so liquid-like neural dynamics—possibly incorporating bidirectional context—are represented as orbits under such flows:

• Each step eliminates non-resonant terms, leaving only those consistent with bidirectional symmetry or the desired invariant manifold.
• The process can be realized analytically within a shrinking domain, or formally in power-series expansions.
• Canonical coordinates thus obtained serve as the natural variables for both forward and backward state propagation, aligning the mathematical architecture with the fundamental bidirectionality of the BiLNN.

5. Symplectic and Noncommutative Generalizations

In the presence of noncommutativity or additional deformations, the structure of BiLNNs is governed by block-matrix symplectic transformations, specified by generalized Bopp's shifts and the normalization of deformed symplectic two-forms (Andrade et al., 2015, Kakuhata et al., 2014). The phase-space variables are mapped via positive, invertible matrices $R$ (constructed from deformed symplectic tensors), and the resulting Hamiltonians acquire explicit dependence on all noncommutative and scaling parameters:

• Bidirectionality is preserved by normalizing the algebra to canonical form, after which standard bidirectional ODE methods (e.g., bi-directional shooting or adjoint equations in machine learning) can be systematically applied.
• Invariant phase-space measures (e.g., $dq\,dp\,\sqrt{g(q)}$ in curved spaces) ensure physical and mathematical consistency for state evolution in both forward and backward temporal directions (Gneiting et al., 2013).

6. Applications and Conceptual Significance

While explicit BiLNN architectures are not instantiated by name in the referenced literature, the outlined algebraic, geometric, and normalization frameworks are directly applicable to the design of such networks. Key implications include:

• Robust normalization procedures for arbitrary phase-space embeddings ensure that both causal and anticausal propagation are consistent with symplectic geometry and, for quantum/physics-inspired architectures, with unitarity.
• Bidirectional, liquid neural computations can leverage the continuous-time canonicalization framework to dynamically adapt to input statistics and temporal correlations, with explicit control over memory, symplectic structure, and dissipation.
• The mathematical analogy extends to physical systems with observer-dependence or coordinate ambiguity, such as the quantization of self-gravitating shells, where changing time-slicings (observer networks) correspond to different bidirectional canonical coordinates and result in distinct quantum representations (Gooding et al., 2019).
• For high-dimensional, noncommutative, or curved-data machine learning problems, BiLNNs constructed in this way are theoretically guaranteed to preserve phase-space volume and normalization under any admissible transformation, facilitating stable and interpretable bidirectional learning.

7. Relation to Broader Research Directions

The confluence of normalization flows, canonical quantization in arbitrary coordinates, and noncommutative/symplectic mapping procedures represents a rigorous foundation for the construction of advanced neural architectures with bidirectional, temporally adaptive computation. These methods underpin current strategies for phase-space aware machine learning (e.g., geometric deep learning, Hamiltonian neural networks) and present a pathway for the principled inclusion of anticausal or future-conditioning information, as required in time-symmetric or reversible dynamical modeling. They also formalize the continuous adaptation of model geometry and dynamics akin to "liquid" state evolution, supporting robust and interpretable inference in complex domains (Treschev, 2023, Gneiting et al., 2013, Blaszak et al., 2013, Andrade et al., 2015, Kakuhata et al., 2014).