Physics-Informed Regularization for Domain-Agnostic Dynamical System Modeling

Learning complex physical dynamics purely from data is challenging due to the intrinsic properties of systems to be satisfied. Incorporating physics-informed priors, such as in Hamiltonian Neural Networks (HNNs), achieves high-precision modeling for energy-conservative systems. However, real-world systems often deviate from strict energy conservation and follow different physical priors.

To address this, we present a framework that achieves high-precision modeling for a wide range of dynamical systems from the numerical aspect, by enforcing Time-Reversal Symmetry (TRS) via a novel regularization term. It helps preserve energies for conservative systems while serving as a strong inductive bias for non-conservative, reversible systems.

While TRS is a domain-specific physical prior, we present the first theoretical proof that TRS loss can universally improve modeling accuracy by minimizing higher-order Taylor terms in ODE integration, which is numerically beneficial to various systems regardless of their properties, even for irreversible systems. By integrating the TRS loss within neural ordinary differential equation models, the proposed model \model demonstrates superior performance on diverse physical systems. It achieves a significant 11.5% MSE improvement in a challenging chaotic triple-pendulum scenario, underscoring \model's broad applicability and effectiveness.

Consider a dynamical system described in the form of $\frac{d\boldsymbol{x}(t)}{dt}=F(\boldsymbol{x}(t))$, where $\boldsymbol{x}(t) \in \Omega$ is the observed states such as positions. The system is said to follow the Time-Reversal Symmetry if there exists a reversing operator $R:\Omega \mapsto \Omega$ such that: $$\frac{d \big( R \circ \boldsymbol{x} (t) \big)}{dt} = -F\big(R \circ \boldsymbol{x}(t)\big),$$ where $\circ$denote the action of functional $R$ on the function $\boldsymbol{x}$.

Theorem 3.1 Let $\Delta t$ denote the integration step size in an ODE solver and $T$ be the prediction length. The reconstruction loss $\mathcal{L}_{pred}$ is $\mathcal{O} (T^3 \Delta t^2)$. The time-reversal loss $\mathcal{L}_{reverse}$ is $\mathcal{O} (T^5 \Delta t^4)$.

We can see two nice properties of our proposed time-reversal loss:

1. Regarding the relationship to $\Delta t$, $\mathcal{L}_{reverse}$ is optimizing a high-order term $\Delta t^4$, which forces the model to predict fine-grained physical properties such as jerk (the derivatives of accelerations). In comparison, the reconstruction loss optimizes $\Delta t^2$, which mainly guides the model to predict the locations/velocities accurately. Therefore, the combined loss enables our model to be more noise-tolerable.
2. Regarding the relationship to $T$, $\mathcal{L}_{reverse}$ is more sensitive to total sequence length ($T^5$), thus it provides more regularization for long-context prediction, a key challenge for dynamic modeling.

To demonstrate the effectiveness of TREAT, We conduct systematic evaluations over five multi-agent systems including three 5-body spring systems, a complex chaotic pendulum system and a real-world motion capture dataset; and four single-agent systems including three spring systems (with only one node) and a chaotic strange attractors system.

Table 1 shows the prediction performance on both multi-agent systems and single-agent systems measured by mean squared error (MSE). We can see that TREAT consistently surpasses other models,highlighting its generalizability and the efficacy of the proposed TRS loss.

We conducted ablation studies on the implementation of $\mathcal{L}_{reverse}$, evaluated performance across different prediction lengths and coefficients, and performed sensitivity analyses on solver choice and observation ratios.

We also demonstrate the chaotic behavior of the triple pendulum which often referred to as the "butterfly effect", and calculate the Maximum Lyapunov Exponent (MLE) to quantify the level of chaos in different kinds of systems.