💻 computer science

Towards trajectory-unsupervised physics-informed neural solvers for molecular dynamics

This paper introduces DINaMo, a physics-informed neural framework that learns molecular dynamics trajectories solely from governing physical laws and initial conditions without relying on any simulator-generated training data, demonstrating the feasibility of recovering physically meaningful short-term dynamics in Lennard-Jones systems.

Original authors: Petros Triantafyllos, Panagiotis Krokidas, Christoforos Rekatsinas

Published 2026-08-10

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CC BY 4.0

Original authors: Petros Triantafyllos, Panagiotis Krokidas, Christoforos Rekatsinas

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where you can watch the invisible dance of atoms, the tiny building blocks that make up everything from the air you breathe to the screen you're reading this on. This is the realm of Molecular Dynamics (MD). Think of it as a high-speed movie camera for the microscopic universe. Scientists use powerful computers to simulate how these atoms move, bounce, and stick together based on the laws of physics. It's like trying to predict the path of every single billiard ball on a table, but instead of a few balls, you have billions, and they are moving so fast that even the best computers struggle to keep up.

The traditional way to do this is like a very careful, step-by-step accountant. The computer calculates where the atoms are at one tiny moment, then uses the laws of physics to figure out where they will be a split-second later, then does it again, and again, and again. It's accurate, but it's incredibly slow and tedious. Because of this, many scientists have turned to Artificial Intelligence (AI) to speed things up. Usually, these AI models are like students who learn by memorizing the answers from a teacher. They are fed massive amounts of data from those slow, traditional simulations and taught to guess the next move. But what if the teacher isn't available? What if we want the AI to learn the rules of the game instead of just memorizing the answers? This is the big question: Can an AI figure out how atoms move just by knowing the laws of physics, without ever seeing a single pre-calculated movie of atoms moving?

This paper introduces a new AI tool called DINaMo (Differentiable Newtonian Molecular Solver) that attempts to answer "yes" to that question. Instead of acting like a student memorizing a textbook, DINaMo acts more like a detective who only has the crime scene's physical laws and the starting position of the suspects. The researchers trained this AI using only the fundamental equations of motion (Newton's laws), the rule that energy must be conserved, and a mathematical description of how atoms push and pull on each other. They didn't feed it any pre-recorded videos of atoms moving, no "correct" answers, and no data from other simulations. They just gave it the starting state and the rules, then asked it to figure out the rest.

The results are promising, though with some caveats. In their tests, DINaMo successfully predicted how a group of 50 to 500 argon atoms would move for a very short period of time. The AI managed to recreate the atoms' positions, their speeds, and even how they clustered together, all without ever seeing a "correct" path to copy. It got the energy right, too, meaning the atoms didn't magically gain or lose energy as they moved, which is a strict requirement of physics. However, the paper notes that this success is currently limited to very short time windows. As the simulation tried to predict further into the future, small errors in the speed of the atoms started to pile up, making the prediction less accurate.

The researchers found that while the AI could handle larger groups of atoms (500 instead of 50) just as well as the smaller groups, stretching the time it had to predict was the real challenge. It's like trying to predict the path of a bouncing ball: you can get it right for a few bounces, but if you try to predict a thousand bounces at once, tiny mistakes in your first guess will eventually throw the whole prediction off. Despite this, the study suggests that it is indeed possible to generate physically meaningful molecular trajectories using only the laws of physics as a teacher, without needing a massive dataset of pre-simulated data. This opens the door to a new kind of AI solver that learns the "why" of motion rather than just the "what," potentially offering a fresh way to tackle complex problems in chemistry and materials science where traditional simulations are too slow and data-hungry.

Technical Summary: Towards Trajectory-Unsupervised Physics-Informed Neural Solvers for Molecular Dynamics

1. Problem Statement

Molecular dynamics (MD) simulations are governed by explicit equations of motion, yet most existing neural approaches designed to accelerate or emulate MD rely heavily on simulator-generated trajectories, forces, or energies for training. This reliance on "trajectory supervision" limits generalization, as models often fail to extrapolate beyond the specific training domain or require extensive datasets that may miss rare high-performing candidates.

The central question addressed by this work is: To what extent can physically meaningful molecular trajectories be recovered directly from the governing physical laws, without using simulator-generated trajectories, forces, velocities, or energies as supervisory targets?

While Physics-Informed Neural Networks (PINNs) offer a framework for training via governing equations rather than data, prior attempts at physics-informed MD solvers (e.g., PND, NP-SNN) have still relied on data-matching terms (comparing predictions to simulator rollouts) or explicit position terms in their loss functions. A fully physics-driven formulation, where trajectories emerge solely from the equations of motion and conservation laws, remains an open challenge.

2. Methodology: DINaMo

The authors introduce DINaMo (Differentiable Newtonian Molecular Solver), a trajectory-unsupervised, physics-informed neural framework. DINaMo represents molecular trajectories as a single continuous, differentiable function of time, $\mathbf{r}_\theta(t)$ , conditioned on an equilibrated initial state.

Core Architecture and Representation

Global Trajectory Representation: Unlike autoregressive time-steppers, DINaMo predicts the entire short-time trajectory window $[0, T]$ as one function. Velocities and accelerations are obtained via exact automatic differentiation of the position function, ensuring kinematic consistency.
Hard Initial-Condition Ansatz: To enforce exact initial conditions ( $\mathbf{r}_0, \mathbf{v}_0, \mathbf{a}_0$ ) without penalty terms, the trajectory is constructed as:
$\mathbf{r}_\theta(t) = \mathbf{r}_0 + t\mathbf{v}_0 + \frac{1}{2}t^2\mathbf{a}_0 + T^2 s^3 \mathbf{g}_\theta(s; \mathbf{r}_0, \mathbf{v}_0)$
where $s = t/T$ is normalized time and $\mathbf{g}_\theta$ is the neural network output. The polynomial baseline captures the analytically known short-time motion (Taylor expansion), allowing the network to focus solely on the nonlinear departure.
Activation Function: The network uses Smooth Inverse Square Root Unit (ISRU) activations, chosen for their well-behaved first and second derivatives, which are critical for computing accelerations and energy rates.

Physics-Only Training Objective

The training objective consists entirely of physics residuals; no simulator data is used as a target. The loss is a hierarchy of terms:

Newton Residual: The primary term minimizes the difference between the predicted acceleration ( $m\ddot{\mathbf{r}}$ ) and the analytic Lennard-Jones force ( $\mathbf{F}(\mathbf{r})$ ).
Conservation Residuals: Momentum and energy conservation act as consistency regularizers.
Augmented-Lagrangian Energy-Rate Constraint: Instead of a fixed penalty, an augmented-Lagrangian term dynamically adjusts the penalty weight to enforce zero energy drift ( $\frac{dE}{dt} = 0$ ), ensuring total energy remains bounded.
Causal Weighting: Residuals are weighted based on the "arrow of time." Large errors in early time steps suppress optimization pressure on later steps until the early trajectory is corrected, preventing the optimizer from fitting late times to an incorrect early state.
Late-Stage Residual Mining: To prevent localized errors from being hidden by global averages, the method applies top- $k$ mining to the largest residuals and emphasizes the tail of the time window in later training stages.

Training Procedure

The solver is trained on a fixed collocation grid aligned with the MD time step. For each initial condition, a separate network is optimized. The method is "trajectory-unsupervised" because it uses only the initial state, the analytic force law, and conservation laws; all subsequent NVE states are withheld during training and used only for evaluation.

3. Key Results

The method was evaluated on Lennard-Jones argon systems in the microcanonical (NVE) ensemble across three experimental settings:

Base Case ( $N=50, \rho=0.15, T=0.12$ ):
- The model achieved high accuracy with position errors $< 5 \times 10^{-5} \sigma$ and velocity errors $< 2 \times 10^{-3} \sigma/\tau_{LJ}$ .
- Total energy drift was comparable to the reference velocity-Verlet integrator.
- The radial distribution function ( $g(r)$ ) was accurately reproduced, preserving structural statistics without being a training target.
Extended Horizon ( $N=50, \rho=0.15, T=0.20$ ):
- Physical coherence (energy conservation and structural statistics) persisted over the longer window.
- However, trajectory accuracy degraded, particularly in velocities (maximum error $\approx 1.57 \times 10^{-1} \sigma/\tau_{LJ}$ ).
- Analysis revealed that kinetic and potential energy errors grew in antiphase, canceling out in the total energy. This indicates the dominant failure mode is a phase mismatch in the energy exchange rather than a loss of conservation.
Dense Scaling ( $N=500, \rho=0.50, T=0.12$ ):
- Contrary to prior data-assisted physics-informed solvers (e.g., PND) which struggled with dense systems, DINaMo maintained high accuracy.
- Maximum position error was $1.18 \times 10^{-5} \sigma$ , and the liquid-like $g(r)$ (with a first-neighbour peak $\approx 3.5$ ) was accurately recovered.
- This demonstrates the method's ability to handle strong many-body force variations and local packing correlations.

4. Significance and Claims

The paper claims that physically meaningful molecular trajectories can emerge directly from physics-only supervision, without learning from precomputed simulator rollouts.

Feasibility of Trajectory-Unsupervised Solvers: The results support the feasibility of neural solvers that rely solely on governing equations (Newton's laws, conservation laws, and analytic potentials) rather than data regression.
Global vs. Local Optimization: By representing the trajectory as a global differentiable function, DINaMo avoids the sequential accumulation of integration errors typical of step-by-step integrators, though it shifts the difficulty to global optimization.
Structural Fidelity: The recovery of the radial distribution function ( $g(r)$ ) and energy conservation without explicit labels suggests the network learns the underlying dynamics rather than merely matching coordinate statistics.

The authors remain modest about the current scope, noting that the method is currently limited to short temporal horizons (up to $T=0.20 \tau_{LJ}$ ) and is instance-specific (a separate model is trained for each initial condition). The work does not claim to have superseded explicit MD integration for long-time production simulations but establishes that the solution to a molecular initial-value problem can be recovered from the initial state and physics alone over nontrivial early-time windows. Future work is identified as extending the physical time via window continuation, amortizing the solver across initial states, and improving derivative accuracy.