Physics-Informed Spiking Neural Networks via Conservative Flux Quantization

TL;DR

A novel Physics-Informed Spiking Neural Network (PISNN) framework is introduced, and to ensure strict physical conservation, the Conservative Leaky Integrate-and-Fire (C-LIF) neuron is designed, whose dynamics structurally guarantee local mass preservation.

Abstract

Real-time, physically-consistent predictions on low-power edge devices is critical for the next generation embodied AI systems, yet it remains a major challenge. Physics-Informed Neural Networks (PINNs) combine data-driven learning with physics-based constraints to ensure the model's predictions are with underlying physical principles.However, PINNs are energy-intensive and struggle to strictly enforce physical conservation laws. Brain-inspired spiking neural networks (SNNs) have emerged as a promising solution for edge computing and real-time processing. However, naively converting PINNs to SNNs degrades physical fidelity and fails to address long-term generalization issues. To this end, this paper introduce a novel Physics-Informed Spiking Neural Network (PISNN) framework. Importantly, to ensure strict physical conservation, we design the Conservative Leaky Integrate-and-Fire (C-LIF) neuron, whose dynamics structurally guarantee local mass preservation. To achieve robust temporal generalization, we introduce a novel Conservative Flux Quantization (CFQ) strategy, which redefines neural spikes as discrete packets of physical flux. Our CFQ learns a time-invariant physical evolution operator, enabling the PISNN to become a general-purpose solver -- conservative-by-construction. Extensive experiments show that our PISNN excels on diverse benchmarks. For both the canonical 1D heat equation and the more challenging 2D Laplace's Equation, it accurately simulates the system dynamics while maintaining perfect mass conservation by design -- a feat that is challenging for conventional PINNs. This work establishes a robust framework for fusing the rigor of scientific computing with the efficiency of neuromorphic engineering, paving the way for complex, long-term, and energy-efficient physics predictions for intelligent systems.

Figures (4)

• Figure 1: Schematic of the PISNN Framework.(a) Conservative Flux Quantization (CFQ). Continuous physical fluxes are discretized into spike events via a learnable 'quota' parameter. This threshold is optimized via distillation from a PINN teacher, bridging continuous physical laws with discrete neural signaling. (b) Network Architecture and C-LIF Dynamics. The network topology is strictly isomorphic to the physical grid, where spatial resolution defines network size ($L_x/d_x$). Each C-LIF neuron directly maps membrane potential to physical quantities, evolving through spike exchange. The architecture supports real-time parameter correction via external sensor data integration.
• Figure 2: Comprehensive validation on the 1D heat equation showing high fidelity, conservation, and efficiency.(a-b) Spatio-temporal evolution of the reference PINN and the zero-shot compiled PISNN. The PISNN rollout is visually indistinguishable from the teacher model. (c) Point-wise error map $(u_{PISNN} - u_{PINN})$, showing negligible deviations. (d) Temporal evolution of RMSE, stabilizing at the order of $10^{-4}$. (e) System-wide mass dynamics. The PISNN trajectory (orange) perfectly tracks the reference (blue), confirming strict conservation adherence. (f) Energy efficiency Pareto frontier. The PISNN (blue curve) allows a tunable trade-off between accuracy and spike count (quota). Notably, it matches PINN accuracy (red star) with a computational cost (spikes) approximately three orders of magnitude lower than the PINN's FLOPs.
• Figure 3: Robust generalization in state space and time.(a-b) Zero-shot generalization to unseen initial conditions. Even when compiled from a teacher trained only on sine waves, the PISNN accurately simulates the diffusion of a Gaussian pulse (a) and the smoothing of a discontinuous step function (b) compared to the FDM reference (Left columns). The low RMSEs (Right columns) confirm the solver's universality. (c) Long-term temporal extrapolation. Left: FDM reference up to $T=2.0$. Middle: The PINN fails to generalize beyond its training horizon ($T_{train}=0.5$, dashed line). Right: Our PISNN maintains stability and accuracy throughout the extended simulation. The RMSE plot quantifies the PINN's divergence versus the PISNN's sustained fidelity.
• Figure 4: Comprehensive validation of the PISNN on the 2D Laplace's equation. (Top rows) Spatio-temporal comparison of the PISNN simulation (left column) against the analytical solution (middle column) at T=0.1s, 0.4s, and 0.8s. The negligible absolute error (right column) demonstrates strong functional simulation fidelity. (Bottom-left) Total mass conservation over time. The PISNN's computed mass (purple line) perfectly matches the analytical mass (blue line), demonstrating perfect physical fidelity and strict adherence to conservation laws. (Bottom-right) The energy efficiency Pareto frontier (Final RMSE vs. Computational Cost). The curve illustrates that the PISNN framework can achieve high accuracy (low RMSE) even at very low computational cost proxies, highlighting its significant energy efficiency potential.