Molecular Augmented Dynamics (MAD)

• Molecular Augmented Dynamics is a modified MD method that augments the Hamiltonian with experimental observables to drive atomistic trajectories toward low-energy configurations.
• It employs machine learning interatomic potentials and differentiable experimental forces to achieve multi-objective optimization in modeling disordered materials.
• MAD formulations utilize linear-scaling strategies for diffraction and local observables, ensuring computational efficiency while matching experimental data such as XRD, ND, and XPS.

Molecular augmented dynamics (MAD) is a modified molecular dynamics method in which the Hamiltonian is augmented by an experimental potential so that atomistic trajectories are driven simultaneously toward low-energy configurations under an interatomic potential and toward agreement with experimental observables. In the literature considered here, MAD is formulated as a multi-objective optimization framework for generating experimentally consistent atomistic structures by design, demonstrated for disordered carbon materials using machine learning interatomic potentials trained from ab-initio data, and subsequently extended with linear-scaling expressions for diffraction and local spectroscopic observables (Zarrouk et al., 23 Aug 2025, Zarrouk et al., 26 Sep 2025). The acronym is not used uniformly across recent atomistic ML research, however, and careful nomenclature is necessary because “MAD” also denotes the Massive Atomistic Diversity dataset in PET-MAD and MAD-1.5, while MAD-SURF expands to Molecular ADsorption on SURFaces (Mazitov et al., 18 Mar 2025, Malosso et al., 2 Mar 2026, Lastre et al., 26 Jan 2026).

1. Definition and nomenclature

In its explicit methodological sense, molecular augmented dynamics is a dynamics-based structure-search procedure that augments standard MD with forces derived from experimental mismatch. The goal is not only to sample thermodynamic ensembles, but to identify structures that simultaneously match multiple experimental observables and exhibit low energies as described by a machine learning interatomic potential trained from ab-initio data (Zarrouk et al., 23 Aug 2025).

Usage	Meaning	Role
MAD	Molecular augmented dynamics	Modified MD with an experimental potential
MAD in PET-MAD and MAD-1.5	Massive Atomistic Diversity	Dataset construction for universal MLIPs
MAD-SURF	Molecular ADsorption on SURFaces	MLIP for adsorption on coinage metal surfaces

This terminological distinction matters because PET-MAD states that MAD in that work refers specifically to the Massive Atomistic Diversity dataset, not to a simulation protocol, and that no alternative or formal definition of “Molecular Augmented Dynamics” or an explicit “MAD” equation is presented beyond this dataset construction (Mazitov et al., 18 Mar 2025). By contrast, the MAD method paper provides an explicit Hamiltonian, explicit experimental forces, and an implementation path in TurboGAP (Zarrouk et al., 23 Aug 2025).

A common misconception is therefore to treat all “MAD” papers as instances of the same formalism. The current literature instead contains at least three distinct usages: a modified-MD method for experiment-constrained structure generation, a family of data-centric universal interatomic-potential resources based on Massive Atomistic Diversity, and acronymic model names such as MAD-SURF (Lastre et al., 26 Jan 2026).

2. Hamiltonian formulation and force decomposition

The central formal modification of MAD is the replacement of the standard Hamiltonian

$\mathcal{H}_{\mathrm{MD}} = T + V$

with

$\mathcal{H} = T + V + \tilde{V},$

where $T$ is kinetic energy, $V$ is interatomic potential energy, and $\tilde{V}$ is an experimental potential that penalizes disagreement between predicted and experimental observables (Zarrouk et al., 23 Aug 2025).

The experimental potential is written as

$$\tilde{V} = \frac{\gamma}{2}\left[\mathbf{w} \odot \left(\mathbf{h}_{\rm pred} - \mathbf{h}_{\rm exp}\right)\right]^2,$$

with $\gamma$ a scaling factor controlling the importance of experimental agreement, $\mathbf{w}$ weights representing importance or inverse uncertainty, $\mathbf{h}_{\rm pred}$ the predicted observable, and $\mathbf{h}_{\rm exp}$ the experimental data (Zarrouk et al., 23 Aug 2025). In this construction, the observable-matching term acts as a bias toward experimentally consistent configurations.

The force on each atom has two contributions,

$\mathcal{H} = T + V + \tilde{V},$0

where $\mathcal{H} = T + V + \tilde{V},$1 is the physical force from $\mathcal{H} = T + V + \tilde{V},$2, and each $\mathcal{H} = T + V + \tilde{V},$3 is an experimental force derived from the gradient of the experimental potential for observable $\mathcal{H} = T + V + \tilde{V},$4 (Zarrouk et al., 23 Aug 2025). For a general observable,

$\mathcal{H} = T + V + \tilde{V},$5

The differentiability requirement is fundamental: the simulated observable must be a differentiable function of atomic positions. This is what allows MAD to include multiple observables simultaneously and, as stated in the method paper, to generalize beyond MD to any structure search or optimization procedure modified with the same experimental potential (Zarrouk et al., 23 Aug 2025).

3. Observable models, differentiability, and linear scaling

The original MAD demonstration used X-ray diffraction (XRD), neutron diffraction (ND), pair distribution function (PDF), and X-ray photoelectron spectroscopy (XPS) data, while emphasizing that the method is general and accepts any experimental observable whose simulated counterpart can be cast as a function of differentiable atomic descriptors (Zarrouk et al., 23 Aug 2025). The accompanying linear-scaling theory paper then derived general equations for MAD with linear-scaling formulations for calculating and matching X-ray/neutron diffraction and local observables, including the core-electron binding energies used in X-ray photoelectron spectroscopy (Zarrouk et al., 26 Sep 2025).

For diffraction, the key computational issue is that the standard Debye equation scales as $\mathcal{H} = T + V + \tilde{V},$6. The linear-scaling MAD formulation replaces that bottleneck with partial pair distribution functions estimated via neighbor lists and kernel density estimators, followed by windowed Fourier transforms in the Ashcroft-Langreth formalism (Zarrouk et al., 26 Sep 2025). The partial PDF is written as

$\mathcal{H} = T + V + \tilde{V},$7

and the corresponding partial structure factors are evaluated up to a finite cutoff $\mathcal{H} = T + V + \tilde{V},$8, which enforces $\mathcal{H} = T + V + \tilde{V},$9 scaling (Zarrouk et al., 26 Sep 2025).

For local scalar observables, the framework is generalized to any quantity differentiable with respect to a local, position-dependent atomic descriptor. The paper’s explicit example is an XPS spectrum,

$T$0

where the predicted binding energy $T$1 depends on local descriptors $T$2 (Zarrouk et al., 26 Sep 2025). The corresponding gradients are obtained analytically by chain rule, which makes these observables compatible with MD integration.

The same work generalized the virial tensor with the experimental forces, enabling generalized barostatting. This allows one to find structures whose density matches that compatible with the experimental observables rather than fixing density a priori (Zarrouk et al., 26 Sep 2025). Scaling tests with TurboGAP demonstrated linear-scaling behavior for both CPU and GPU implementations, with the GPU implementation giving a 100$T$3 speedup compared to the CPU (Zarrouk et al., 26 Sep 2025).

4. Interatomic potentials, workflow, and physical realism

MAD relies on an interatomic potential $T$4 that is accurate enough to prevent the structure search from converging to experimentally compatible but physically implausible configurations. In the main demonstration, the interatomic potentials were machine learning potentials, specifically Gaussian Approximation Potentials (GAPs), used for accurate modeling of the potential energy surface at computational cost similar to empirical force fields (Zarrouk et al., 23 Aug 2025). The same source states that this removes the need for artificial connectivity or bonding constraints and enables efficient large-scale atomistic simulations involving thousands of atoms.

The operational workflow is an annealing-based MD protocol. Initial random structures are annealed, for example via a melt-and-quench protocol, while MAD minimizes a multi-objective function comprising both the physical energy and the deviation from experimental observable(s) (Zarrouk et al., 23 Aug 2025). Simulations may be run in NVT or NPT, and under NPT the modified virial tensor allows the experimental density to emerge from matching (Zarrouk et al., 23 Aug 2025).

A critical methodological point is that the experimental bias is not intended to replace physical energetics. Rather, it supplements the interatomic potential so that structures remain low-energy while being driven toward experimental consistency. The method paper explicitly frames this as an improvement over earlier RMC/HRMC-type approaches, which could fit experimental data at the cost of realism (Zarrouk et al., 23 Aug 2025). The linear-scaling theory paper makes a related point from the algorithmic side, noting that prior approaches were hindered by poor observable gradient accuracy, computational inefficiency, $T$5 scaling, or inability to include complex local-probe observables (Zarrouk et al., 26 Sep 2025).

Another important procedural detail is that, after removing $T$6, the system remains in a low-energy and experimentally compatible state (Zarrouk et al., 23 Aug 2025). This indicates that the method is not merely producing transiently biased configurations, but can locate metastable structures compatible with the chosen observables.

5. Demonstrated systems and outcomes

The initial MAD paper demonstrated feasibility on several carbon materials: glassy carbon, nanoporous carbon, ta-C, a-C:D, and a-CO$T$7, matching their respective X-ray diffraction, neutron diffraction, pair distribution function, and X-ray photoelectron spectroscopy data using the same initial structure and underlying MLP (Zarrouk et al., 23 Aug 2025). The stated objective was to identify representative structures of these disordered materials by direct optimization against experiment rather than by unconstrained sampling and retrospective selection.

The later linear-scaling paper sharpened the interpretation of these outcomes. It states that MAD simulations can both find metastable structures compatible with non-equilibrium experimental synthesis and lower energy structures than alternative computational sampling protocols, like the melt-quench approach (Zarrouk et al., 26 Sep 2025). This is an important distinction: when experimental synthesis is non-equilibrium, agreement with experiment may legitimately require metastable rather than minimum-energy structures, and MAD is formulated to capture that trade-off through the parameter $T$8.

These results also clarify what MAD is not. It is not simply a method for accelerating equilibrium MD, nor is it only an inverse-fitting procedure akin to reverse Monte Carlo. Its defining feature is simultaneous optimization of interatomic potential energy and experimental agreement under MD dynamics (Zarrouk et al., 23 Aug 2025, Zarrouk et al., 26 Sep 2025). The phrase from the original abstract that the method enables a computational “microscope” into experimental structures is therefore best understood as a claim about interpretable structure generation under experimental constraints, not about direct reconstruction from raw measurements alone (Zarrouk et al., 23 Aug 2025).

6. Adjacent MAD ecosystems and broader research directions

Recent papers extend the practical landscape around MAD, although not always in the same formal sense. PET-MAD is a generally applicable machine-learning interatomic potential trained on the Massive Atomistic Diversity dataset, comprising 95,595 structures and 85 elements; it is intended to deliver near-quantitative, out-of-the-box accuracy across organic and inorganic materials, and PET-MAD can be efficiently fine-tuned with LoRA to deliver full quantum mechanical accuracy with a minimal number of targeted calculations (Mazitov et al., 18 Mar 2025). In that work, “MAD” refers to dataset construction rather than a modified-Hamiltonian MD protocol.

MAD-1.5 extends the original MAD dataset to 216,803 structures and 102 elements, using a single, standardized all-electron DFT workflow with the $T$9SCAN meta-GGA functional and outlier removal using Last-Layer Prediction Rigidity (LLPR) uncertainty (Malosso et al., 2 Mar 2026). The associated PET-MAD-1.5 models are presented as generally applicable $V$0SCAN interatomic potentials with benchmark accuracy and stability in challenging simulation protocols (Malosso et al., 2 Mar 2026). This suggests a data-centric route for strengthening the interatomic potential component $V$1 in future MAD simulations.

MAD-SURF is a machine learning interatomic potential specifically tailored for molecular adsorption on Cu(111), Ag(111), and Au(111), trained on over 77,000 DFT-calculated configurations spanning adsorption structures, ab initio molecular dynamics, normal mode sampling, and non-covalent aggregates (Lastre et al., 26 Jan 2026). It achieves DFT-comparable accuracy while enabling simulations orders of magnitude faster than density functional theory and has been demonstrated on experimentally characterized systems including organic monolayers, polycyclic aggregates, flexible biomolecules, and the long-range herringbone reconstruction of gold (Lastre et al., 26 Jan 2026). Although MAD-SURF is not the experimental-potential formalism, it exemplifies the kind of specialized MLIP infrastructure that can support augmented atomistic workflows.

A different but related direction is represented by MD-LLM-1, described as the first LLM framework specifically adapted for molecular dynamics trajectory generation. Fine-tuned from Mistral 7B, it can learn protein dynamics and generate conformations characteristic of states not included in the training data, as shown for T4 lysozyme and Mad2 (Murtada et al., 21 Jul 2025). The same paper states that the model does not directly model thermodynamics or kinetics. A plausible implication is that “augmentation” in atomistic simulation is now being pursued along three complementary axes: experimental-force augmentation of dynamics, dataset-driven augmentation of transferable interatomic potentials, and generative augmentation of conformational exploration (Murtada et al., 21 Jul 2025, Mazitov et al., 18 Mar 2025).

Across these strands, the most stable formal meaning of molecular augmented dynamics remains the modified-MD method defined by

$V$2

with differentiable experimental observables supplying additional forces during atomistic evolution (Zarrouk et al., 23 Aug 2025). The surrounding ecosystem of MAD-named datasets, MLIPs, and generative models indicates an expanding research program centered on bringing experimental consistency, transferability, and efficient exploration into the core of atomistic simulation.