Equivariant Graph Neural Networks (EGNNs) have demonstrated significant success in modeling microscale systems, including those in chemistry, biology and materials science.

In this section, we provide a brief introduction to the concepts from Group Theory which we need in our derivations. A group is a pair (G,)containing a set Gand a binary operation: G G! G,(h,g) 7! h g which satisfies the group axioms: Associativity: 8a,b,c 2 Ga (b c)=( a b) c Identity: 9e 2 G: 8g 2 Gg e = e g = g Inverse: 8g 2 G 9g 1 2 G: g g 1 = g 1 g = e The operation is the group law of G. The inverse elements g 1 of an element g, and the identity element e are unique. In addition, if the group law is also commutative, the group G is an abelian group. To simplify the notation, we commonly write ab instead of a b. It is also common to denote the group (G,) just with the name of its underlying set G. The order of a group G is the cardinality of its set and is indicated by |G|. A group G is finite when |G|2 N, i.e., when it has a finite number of elements. A compact group is a group that is also a compact topological space with continuous group operation. Given a group G, its action on a set X is a map . A simple example of group action is the group law itself: G G! Gwhich defines an action of G on its own elements (X = G). Another important action is the one defined on signals overs the group G. Given a signal x: G! R, the action of an element g 2 G maps x 7! g.x, [g.x](h):= x(g 1h).

In this section we briefly introduce the representation theory of the three groups we used in this work. Planar rotations group SO(2) The standard representation of r 2 SO(2) is as a 2 2 rotation matrix (r)= cos sin sin cos The complex irreducible representations are often used and correspond to the circular harmonics. Planar rotations and reflections group O(2) The standard representation of O(2) is as a 2 2 orthogonal matrix (r)= cos sin sin cos and (r f)= cos sin sin cos 10 01 Apart from the trivial representation 0,0(h)=1 8h 2 O(2) and the sign-flip representation 1,0(r)=1 and 1,0(f)= 1, all other irreps are 2 dimensional. These representations are isomorphic to the Wigner D matrices. In particular, 0 is the trivial representation and i is isomorphic to the standard representation of SO(3) as 3 3 rotation matrices. An element g =( m,r) 2 O(3) is a pair of a mirroring m 2{ e,mz} and a rotation r 2 SO(3). In general, if G is a group, we denote with bG the set of its irreducible representations. Recall the generative process for cryo-EM images: oi = (g 1i) with gi 2 SO(3) (12) 14 Let Rz = SO(2) < SO(3) the subgroup of SO(3) containing rotations around the Z axis and H = O(2) < SO(3) the subgroup containing also the rotation ry by around the Y axis.

Fighting with thecurse of dimensionalityby leveraging low-dimensional intrinsic structures has become an important guiding principle in modern data science.

B: Overview of the proposed PhiSNet architecture.