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 @@@@1@@@@ (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 $k$ is the wave number. The network operates on $N$ discretized values $0 < \rho_1 < \cdots < \rho_N$ ($\rho_N = k r_{\text{max}}$). At each $\rho_n$, the forward and backward hidden states, $h^{(f)}_n, h^{(b)}_n \in \mathbb{R}^D$, are updated according to:

$$\begin{align*} g^{(f)}_n & = \sigma(W_g [h^{(f)}_{n-1}; x_n] + b_g) \in \mathbb{R}^D, \ c^{(f)}_n & = \tanh(W_c [h^{(f)}_{n-1}; x_n] + b_c) \in \mathbb{R}^D, \ h^{(f)}_n & = \exp(-g^{(f)}_n \Delta \rho) \odot h^{(f)}_{n-1} + [1 - \exp(-g^{(f)}_n \Delta \rho)] \odot c^{(f)}_n, \end{align*}$$

with an analogous update for $h^{(b)}_n$ propagating in the reverse direction. The concatenated hidden state $h_n = [h^{(f)}_n; h^{(b)}_n] \in \mathbb{R}^{2D}$ feeds into a fully connected combiner and decoder, yielding real and imaginary wave function components

$$y_n = [\psi_R(\rho_n);\, \psi_I(\rho_n)] = V \cdot \mathrm{ReLU}(U h_n + d) + e,$$

where $U$, $V$ are learned weights, and $d$, $e$ are biases. All operations are differentiable by construction.

3. Feature Encoding and Physics-Informed Inputs

Each spatial point $r_n$ is associated with a nine-dimensional feature vector: $$x_n = \left( \rho_n / \rho_{\max},\; V_R(r_n)/E,\; W(r_n)/E,\; \eta/\eta_{\max}, \; \sin\varphi_{\mathrm{WKB}}(r_n),\; \cos\varphi_{\mathrm{WKB}}(r_n),\; D(r_n),\; l/l_{\max},\; A/A_{\max} \right)^\top,$$ where:

• $V_R(r), W(r)$ are real and imaginary parts of the local optical potential (scaled by projectile energy $E$),
• $\eta$ is the Sommerfeld parameter,
• $\varphi_{\mathrm{WKB}}(r)$ is the accumulated semiclassical phase $\int_0^r k_{\text{local}}(r')\, dr'$,
• $$D(r) = \exp[-\int_0^r \operatorname{Im}\, k_{\text{local}}(r')\, dr']$$ is a semiclassical absorption factor,
• $l$ is the partial wave, and $A$ 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 $D=128$ liquid neurons per direction, exploiting approximately 50% sparse connectivity, with a total parameter count near $1.2 \times 10^6$. Training utilizes approximately $74\,400$ Numerov-computed solutions spanning 12 nuclei ($12 \leq A \leq 208$), $l=0 \ldots 30$, and $E \in [1,200]$ MeV for both protons and neutrons, discretized to $T=100$ spatial points per wave function.

The objective function is mean-squared error over all spatial points and samples: $$\mathcal{L} = \frac{1}{N T} \sum_{i=1}^N \sum_{n=1}^T \left[ (\psi_R^{\rm pred}(\rho_n) - \psi_R^{\rm true}(\rho_n))^2 + (\psi_I^{\rm pred}(\rho_n) - \psi_I^{\rm true}(\rho_n))^2 \right],$$ 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: $$(\text{optical potential},\,E,\,l,\,A) \mapsto (u_l(r)_{\mathrm{Re}},\,u_l(r)_{\mathrm{Im}})$$, supporting automatic gradient computation essential for downstream optimization and uncertainty propagation.

5. Phase-Space Coordinate Normalization and Generalization Principle

BiLNN’s phase-space normalization, $\rho = k r$, ensures that oscillatory structure induced by the de Broglie wavelength $\lambda = 2\pi/k$ is mapped to a universal period in $\rho$-space ($2\pi$ for all $E$). 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 $E=1\,\text{MeV}$ to $200\,\text{MeV}$—a $4.5\times$ variation in $\lambda$ if operating in $r$-space.

6. Accuracy, Physical Observables, and Generalization Performance

On held-out data, BiLNN achieves a root-mean-square relative wave function error of $1.2\%$ overall. Error rates by partial wave $l$ are approximately $1.0\%$ over $l=8$–$16$, rising to $1.8\%$ at $l\approx 0$ and $l\approx 30$. Across targets, errors remain within $0.8$–$1.3\%$ for $A\leq 120$ and up to $1.8\%$ for the heaviest nuclei ($A=197,208$). For projectile energies $E < 20\,\text{MeV}$, errors are $\sim 4\%$, dropping to $\sim 1\%$ for $E > 100\,\text{MeV}$.

Physical observables computed from the predicted wave functions include elastic $S$-matrix elements: $$f_l = -\frac{1}{E} \int_0^\infty F_l(\eta, kr) V(r) \psi_l(r)\, dr, \quad S_l = 1 + 2ik f_l,$$ as well as elastic scattering cross sections $d\sigma/d\sigma_{\text{Ruth}}$ for protons and $d\sigma/d\Omega$ 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 $\mathcal{O}(1\!-\!2\%)$.

Crucially, BiLNN successfully extrapolates to nuclei excluded from training (e.g., $^{24}$Mg, $^{63}$Cu, $^{184}$W), maintaining wave function errors $\lesssim 1.5\%$ and observable fidelity comparable to in-sample targets. An ablation replacing bidirectional recurrence with a forward-only liquid layer increases wave function error from $1.2\%$ to $1.4\%$ (approximately $14\%$ relative degradation), especially for high $l$ 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 $A,Z$ 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).