Bidirectional Liquid Neural Networks (BiLNN)

• BiLNN is a continuous-time recurrent architecture that fuses learnable ODE dynamics with bidirectional propagation to enforce both Dirichlet and asymptotic boundary conditions.
• It maps complex optical potential parameterizations to nuclear scattering wave functions, achieving sub‑percent error across diverse energies and nuclear species.
• The differentiable design supports gradient‑based optimization and uncertainty quantification, serving as a robust surrogate in nuclear data evaluation.

A Bidirectional Liquid Neural Network (BiLNN) is a class of continuous-time recurrent neural architectures designed for the differentiable emulation of physical boundary-value problems, exemplified by its application to global nucleon-nucleus optical model calculations. BiLNNs synthesize liquid (continuous-time ODE-driven) recurrence and bidirectional propagation to provide a mapping from complex optical potential parameterizations to scattering wave functions, while satisfying physical boundary conditions and preserving analytical differentiability. The architecture enables gradient-based optimization and uncertainty quantification in nuclear modeling, producing observables with sub-percent error and demonstrating transferability across a broad parameter space, including extrapolation to untrained nuclear species (Lei, 27 Dec 2025).

1. Architectural Foundations and Relationship to Liquid/Reservoir Computing

The BiLNN architecture generalizes reservoir computing by employing learnable continuous-time ordinary differential equation (ODE) dynamics and by enforcing bidirectional recurrence tailored to physics boundary-value problems. In contrast to discrete-time gated recurrent units (LSTM/GRU), the BiLNN hidden state $h(r)$ evolves according to a first-order ODE: $$\frac{d h(r)}{d r} = -g(r) \odot h(r) + [1 - g(r)] \odot c(r),$$ where $g(r)$ is a learned leak gate and $c(r)$ is a learned candidate drive. This ODE form admits a closed-form solution over each step $\Delta r$, mitigating vanishing/exploding gradient pathologies on long sequences.

Bidirectionality is enforced by running two parallel liquid layers: one propagates forward from $r=0$ (origin) to $r_{\text{max}}$ (asymptotic region), while the other propagates backward from $r_{\text{max}}$ to $r=0$. At each spatial position, the hidden states from both passes are merged, ensuring explicit conditioning on both Dirichlet and asymptotic boundary conditions. This design is particularly well-suited to radial Schrödinger problems, where both boundary behaviors are formally required.

2. Mathematical Formulation and Internal Dynamics

Spatial coordinates are mapped to a dimensionless phase-space form: $\rho = k r$, where $$\frac{d h(r)}{d r} = -g(r) \odot h(r) + [1 - g(r)] \odot c(r),$$0 is the wave number. The network operates on $$\frac{d h(r)}{d r} = -g(r) \odot h(r) + [1 - g(r)] \odot c(r),$$1 discretized values $$\frac{d h(r)}{d r} = -g(r) \odot h(r) + [1 - g(r)] \odot c(r),$$2 ($$\frac{d h(r)}{d r} = -g(r) \odot h(r) + [1 - g(r)] \odot c(r),$$3). At each $$\frac{d h(r)}{d r} = -g(r) \odot h(r) + [1 - g(r)] \odot c(r),$$4, the forward and backward hidden states, $$\frac{d h(r)}{d r} = -g(r) \odot h(r) + [1 - g(r)] \odot c(r),$$5, are updated according to:

$$\frac{d h(r)}{d r} = -g(r) \odot h(r) + [1 - g(r)] \odot c(r),$$6

with an analogous update for $$\frac{d h(r)}{d r} = -g(r) \odot h(r) + [1 - g(r)] \odot c(r),$$7 propagating in the reverse direction. The concatenated hidden state $$\frac{d h(r)}{d r} = -g(r) \odot h(r) + [1 - g(r)] \odot c(r),$$8 feeds into a fully connected combiner and decoder, yielding real and imaginary wave function components

$$\frac{d h(r)}{d r} = -g(r) \odot h(r) + [1 - g(r)] \odot c(r),$$9

where $g(r)$0, $g(r)$1 are learned weights, and $g(r)$2, $g(r)$3 are biases. All operations are differentiable by construction.

3. Feature Encoding and Physics-Informed Inputs

Each spatial point $g(r)$4 is associated with a nine-dimensional feature vector: $g(r)$5 where:

• $g(r)$6 are real and imaginary parts of the local optical potential (scaled by projectile energy $g(r)$7),
• $g(r)$8 is the Sommerfeld parameter,
• $g(r)$9 is the accumulated semiclassical phase $c(r)$0,
• $c(r)$1 is a semiclassical absorption factor,
• $c(r)$2 is the partial wave, and $c(r)$3 the target mass, normalized to their maxima.

All features are pre-normalized and processed through a two-layer encoder MLP (ReLU activations), yielding the high-dimensional representation used in the liquid layers.

4. Network Training, Parameterization, and Differentiability

The BiLNN as implemented employs $c(r)$4 liquid neurons per direction, exploiting approximately 50% sparse connectivity, with a total parameter count near $c(r)$5. Training utilizes approximately $c(r)$6 Numerov-computed solutions spanning 12 nuclei ($c(r)$7), $c(r)$8, and $c(r)$9 MeV for both protons and neutrons, discretized to $\Delta r$0 spatial points per wave function.

The objective function is mean-squared error over all spatial points and samples: $\Delta r$1 optimized using AdamW for 500 epochs with weight decay regularization. Training converges within 3–4 hours on a single GPU. The completed model acts as a fully analytic, differentiable surrogate: $\Delta r$2, supporting automatic gradient computation essential for downstream optimization and uncertainty propagation.

5. Phase-Space Coordinate Normalization and Generalization Principle

BiLNN’s phase-space normalization, $\Delta r$3, ensures that oscillatory structure induced by the de Broglie wavelength $\Delta r$4 is mapped to a universal period in $\Delta r$5-space ($\Delta r$6 for all $\Delta r$7). This eliminates the need for the network to learn a priori a continuum of energy-dependent wavelengths, simplifying the learning task and enabling a single model to generalize across $\Delta r$8 to $\Delta r$9—a $r=0$0 variation in $r=0$1 if operating in $r=0$2-space.

6. Accuracy, Physical Observables, and Generalization Performance

On held-out data, BiLNN achieves a root-mean-square relative wave function error of $r=0$3 overall. Error rates by partial wave $r=0$4 are approximately $r=0$5 over $r=0$6–$r=0$7, rising to $r=0$8 at $r=0$9 and $r_{\text{max}}$0. Across targets, errors remain within $r_{\text{max}}$1–$r_{\text{max}}$2 for $r_{\text{max}}$3 and up to $r_{\text{max}}$4 for the heaviest nuclei ($r_{\text{max}}$5). For projectile energies $r_{\text{max}}$6, errors are $r_{\text{max}}$7, dropping to $r_{\text{max}}$8 for $r_{\text{max}}$9.

Physical observables computed from the predicted wave functions include elastic $r_{\text{max}}$0-matrix elements: $r_{\text{max}}$1 as well as elastic scattering cross sections $r_{\text{max}}$2 for protons and $r_{\text{max}}$3 for neutrons. The model reliably recovers diffraction minima spanning four orders of magnitude in cross section, with a root-mean-square cross-section error of $r_{\text{max}}$4.

Crucially, BiLNN successfully extrapolates to nuclei excluded from training (e.g., $r_{\text{max}}$5Mg, $r_{\text{max}}$6Cu, $r_{\text{max}}$7W), maintaining wave function errors $r_{\text{max}}$8 and observable fidelity comparable to in-sample targets. An ablation replacing bidirectional recurrence with a forward-only liquid layer increases wave function error from $r_{\text{max}}$9 to $r=0$0 (approximately $r=0$1 relative degradation), especially for high $r=0$2 and near the domain boundary, substantiating the architectural necessity for bidirectionality in enforcing dual boundary conditions.

7. Applications and Significance

BiLNN affords a differentiable, physics-informed surrogate for nuclear wave function computation, facilitating gradient-based optical-model parameter optimization and uncertainty quantification. Its design, grounded in phase-space normalization and bidirectional, ODE-driven recurrence, generalizes across a broad spectrum of projectile energies, partial waves, and nuclear targets. The demonstrated extrapolation performance suggests BiLNN has internalized the smooth $r=0$3 dependence typical of global optical potentials (e.g., KD02), rather than simply memorizing specific cases. These properties make BiLNN a compelling candidate for integration into modern nuclear data evaluation pipelines, where rapid, differentiable, and physically accurate surrogate models are increasingly vital (Lei, 27 Dec 2025).