diff --git a/c10/core/DispatchKey.cpp b/c10/core/DispatchKey.cpp
index 775e47ca99d..bffafc59168 100644
--- a/c10/core/DispatchKey.cpp
+++ b/c10/core/DispatchKey.cpp
@@ -206,7 +206,7 @@ const char* toString(DispatchKey t) {
     switch (bc) {                                  \
       C10_FORALL_BACKEND_COMPONENTS(ENTRY, prefix) \
       default:                                     \
-        return #prefix "Unknown";                  \
+        return #prefix "Undefined";                \
     }
 
         C10_FORALL_FUNCTIONALITY_KEYS(FORALL_BC)
diff --git a/functorch/_src/aot_autograd.py b/functorch/_src/aot_autograd.py
index 5294399dbca..de8a00c68f6 100644
--- a/functorch/_src/aot_autograd.py
+++ b/functorch/_src/aot_autograd.py
@@ -278,6 +278,7 @@ class AOTConfig:
     bw_compiler: Callable
     partition_fn: Callable
     decompositions: Dict[Callable, Callable]
+    num_params_buffers: int
 
 
 def aot_dispatch_base(flat_fn, flat_args: List[Tensor], aot_config: AOTConfig):
@@ -491,6 +492,11 @@ def create_aot_dispatcher_function(
 
     The resulting compiled forward and backward graphs are then wrapped up in a
     ``torch.autograd.Function`` object.
+
+    The calling convention here is that the first aot_config.num_params_buffers
+    inputs in flat_args are parameters and buffers, and the rest are inputs.
+
+    We use this to assume that parameters/buffer's shapes don't change.
     """
 
     # This is the main entry point.
@@ -514,19 +520,26 @@ def create_aot_dispatcher_function(
         # coordinate flags
         config.use_fake_tensor = False
 
-    fake_mode = FakeTensorMode() if config.use_fake_tensor else nullcontext()
+    if config.use_dynamic_shapes:
+        assert config.use_fake_tensor, "Dynamic shapes only works with fake tensor"
+
+    shape_env = ShapeEnv() if config.use_dynamic_shapes else None
+    fake_mode = FakeTensorMode(shape_env=shape_env) if config.use_fake_tensor else nullcontext()
     cross_ref = CrossRefFakeMode() if config.debug_fake_cross_ref else nullcontext()
     python_dispatcher_mode = enable_python_dispatcher() if config.use_dynamic_shapes else nullcontext()
-    shape_env = ShapeEnv() if config.use_dynamic_shapes else None
 
     with torch.autograd.set_multithreading_enabled(False), preserve_rng_state(), cross_ref, fake_mode, python_dispatcher_mode:
 
         def process_inputs(flat_args):
             if config.use_fake_tensor:
-                def convert(x):
-                    return fake_mode.from_tensor(x, shape_env=shape_env)
+                def convert(idx, x):
+                    if not isinstance(x, torch.Tensor):
+                        return x
+                    if idx < aot_config.num_params_buffers and config.static_weight_shapes:
+                        return fake_mode.from_tensor(x, static_shapes=True)
+                    return fake_mode.from_tensor(x, static_shapes=False)
 
-                return pytree.tree_map_only(Tensor, convert, flat_args)
+                return [convert(idx, x) for idx, x in enumerate(flat_args)]
             else:
                 return flat_args
 
@@ -587,6 +600,7 @@ def aot_function(
     bw_compiler: Optional[Callable] = None,
     partition_fn: Callable = default_partition,
     decompositions: Optional[Dict] = None,
+    num_params_buffers: int = 0,
     hasher_type=None,  # deprecated
     static_argnums: Optional[Tuple[int]] = None,  # deprecated
 ) -> Callable:
@@ -650,6 +664,7 @@ def aot_function(
         bw_compiler=bw_compiler,
         partition_fn=partition_fn,
         decompositions=decompositions,
+        num_params_buffers=num_params_buffers,
     )
     cached_res = None
 
@@ -734,7 +749,10 @@ def aot_module(mod: nn.Module, *args, **kwargs) -> nn.Module:
         params_and_buffers = {**named_params, **named_buffers}
         return stateless.functional_call(mod, params_and_buffers, args, kwargs)
 
-    compiled_f = aot_function(functional_call, *args, **kwargs)
+    named_params = dict(_named_parameters(mod, remove_duplicate=False))
+    named_buffers = dict(_named_buffers(mod, remove_duplicate=False))
+    num_params_buffers = len(named_params) + len(named_buffers)
+    compiled_f = aot_function(functional_call, num_params_buffers=num_params_buffers, *args, **kwargs)
 
     class AOTModule(nn.Module):
         def __init__(self):
@@ -743,8 +761,8 @@ def aot_module(mod: nn.Module, *args, **kwargs) -> nn.Module:
 
         def forward(self, *args, **kwargs):
             return compiled_f(
-                dict(_named_parameters(mod, remove_duplicate=False)),
-                dict(_named_buffers(mod, remove_duplicate=False)),
+                named_params,
+                named_buffers,
                 *args,
                 **kwargs,
             )
@@ -812,6 +830,7 @@ def aot_module_simplified(mod: nn.Module, *top_args, **top_kwargs) -> nn.Module:
             bw_compiler=bw_compiler,
             partition_fn=partition_fn,
             decompositions=decompositions,
+            num_params_buffers=params_len,
         )
 
         compiled_fn = None
diff --git a/functorch/_src/config.py b/functorch/_src/config.py
index e473d1129ea..2dacdd38fa3 100644
--- a/functorch/_src/config.py
+++ b/functorch/_src/config.py
@@ -23,3 +23,5 @@ debug_graphs = os.environ.get('AOT_FX_GRAPHS', False)
 debug_joint = os.environ.get('AOT_FX_GRAPHS_JOINT', False)
 
 use_dynamic_shapes = os.getenv('AOT_DYNAMIC_SHAPES', False)
+
+static_weight_shapes = True
diff --git a/test/functorch/test_aotdispatch.py b/test/functorch/test_aotdispatch.py
index 8d1c0dba701..df25f90e55f 100644
--- a/test/functorch/test_aotdispatch.py
+++ b/test/functorch/test_aotdispatch.py
@@ -1060,6 +1060,7 @@ symbolic_aot_autograd_failures = {
     xfail('nn.functional.avg_pool1d', ''),  # Cannot call sizes() on tensor with symbolic sizes/strides
     xfail('nn.functional.avg_pool2d', ''),  # aten.avg_pool2d.default - couldn't find symbolic meta function/...
     xfail('nn.functional.avg_pool3d', ''),  # aten.avg_pool3d.default - couldn't find symbolic meta function/...
+    skip('nn.functional.batch_norm', ''),  # '0 is not tracked with proxy for <torch.fx.experimental.proxy_te..
     xfail('nn.functional.bilinear', ''),  # Cannot call sizes() on tensor with symbolic sizes/strides
     xfail('nn.functional.binary_cross_entropy', ''),  # aten.fill_.Scalar - couldn't find symbolic meta funct...
     xfail('nn.functional.conv1d', ''),  # Cannot call sizes() on tensor with symbolic sizes/strides
diff --git a/test/test_dynamic_shapes.py b/test/test_dynamic_shapes.py
index 0f12734ffd6..87b1dd9aa82 100644
--- a/test/test_dynamic_shapes.py
+++ b/test/test_dynamic_shapes.py
@@ -113,13 +113,14 @@ class FakeSymbolicTensor(torch.Tensor):
 
 
 def create_symbolic_tensor(name, arg, shape_env, storage_offset=0):
-    sym_shapes = tuple([shape_env.create_symint(f"{name}_{idx}", val) for idx, val in enumerate(arg.size())])
-    sym_strides = tuple([shape_env.create_symint(f"{name}_{idx}_stride", val) for idx, val in enumerate(arg.stride())])
+    sym_shapes, sym_strides = shape_env.create_symbolic_sizes_strides(arg)
     return FakeSymbolicTensor(sym_shapes, sym_strides, arg.dtype, arg.layout, arg.requires_grad, arg.device, storage_offset)
 
 
 CPP_SYMINT_CLASS = type(torch.SymIntNode.new_symint(1))
 
+def create_symint(shape_env, i):
+    return shape_env.create_symintnode(shape_env.create_symbol(i))
 
 @skipIfTorchDynamo("Creating ShapeEnv fails for confusing reasons (also we never expect dynamo to see code like this)")
 class TestPySymInt(TestCase):
@@ -128,8 +129,8 @@ class TestPySymInt(TestCase):
     def test_arith_ops(self):
         shape_env = ShapeEnv()
         symints = []
-        for i in range(5):
-            symints.append((i, shape_env.create_symint(f"s{i}", i)))
+        for i in range(2, 5):
+            symints.append((i, create_symint(shape_env, i)))
 
         ops = [operator.add, operator.sub, operator.floordiv, operator.mul, operator.mod]
 
@@ -143,10 +144,10 @@ class TestPySymInt(TestCase):
     def test_reverse_arith_ops(self):
         shape_env = ShapeEnv()
 
-        a = shape_env.create_symint("s1", 2)
+        a = create_symint(shape_env, 2)
         self.assertTrue(5 // a == 5 // 2)
 
-        a = shape_env.create_symint("s1", 2)
+        a = create_symint(shape_env, 2)
         self.assertTrue(5 * a == 5 * 2)
 
 
@@ -172,7 +173,7 @@ class TestPySymInt(TestCase):
         self.assertTrue(x.size(2) == 3)
         self.assertTrue(isinstance(x.size(2), CPP_SYMINT_CLASS))
 
-        offset = shape_env.create_symint("offset", 2)
+        offset = create_symint(shape_env, 2)
         y = create_symbolic_tensor("x", torch.randn(5, 4, 3), shape_env, offset)
         self.assertTrue(isinstance(y.storage_offset(), CPP_SYMINT_CLASS))
         self.assertTrue(y.storage_offset() == 2)
@@ -207,7 +208,7 @@ class TestPySymInt(TestCase):
         y = create_symbolic_tensor("y", torch.randn(5, 4, 1), shape_env)
         LAST_DIM = 2
         z = x.narrow_copy(LAST_DIM, 0, y.shape[LAST_DIM])
-        self.assertTrue(z.shape[2] == int(y.shape[2]))
+        self.assertTrue(z.shape[2] == y.shape[2])
 
         # arithmetic expr with two symints
         z = x.narrow_copy(LAST_DIM, 0, x.shape[LAST_DIM] - y.shape[LAST_DIM])
@@ -317,28 +318,28 @@ class TestPySymInt(TestCase):
     @skipIfNoSympy
     def test_meta_symint(self):
         shape_env = ShapeEnv()
-        a0 = shape_env.create_symint("a0", 2)
+        a0 = create_symint(shape_env, 2)
         r = torch.empty(a0, device='meta')
         self.assertIsInstance(r.shape[0], CPP_SYMINT_CLASS)
 
     @skipIfNoSympy
     def test_guard_int(self):
         shape_env = ShapeEnv()
-        a0 = shape_env.create_symint("a0", 2)
+        a0 = create_symint(shape_env, 2)
         self.assertEqual(a0.guard_int(), 2)
-        self.assertEqual(str(shape_env.guards[0][0]), "a0")
+        self.assertEqual(str(shape_env.guards[0][0]), "s0")
         self.assertEqual(shape_env.guards[0][1], 2)
 
     @skipIfNoSympy
     def test_int_conversion(self):
         shape_env = ShapeEnv()
-        a0 = shape_env.create_symint("a0", 2)
+        a0 = create_symint(shape_env, 2)
         self.assertRaisesRegex(RuntimeError, "Trying to extract", lambda: int(a0))
 
     @skipIfNoSympy
     def test_symint_as_scalar(self):
         shape_env = ShapeEnv()
-        a0 = shape_env.create_symint("a0", 2)
+        a0 = create_symint(shape_env, 2)
 
         sym_int_encountered = False
 
@@ -372,18 +373,18 @@ class TestPySymInt(TestCase):
 
         self.assertExpectedInline(mock_stdout.getvalue().strip(), """\
 class f(torch.nn.Module):
-    def forward(self, a_1: f32[t0.size(0),t0.size(1)], b_1: f32[t1.size(0),t0.size(1)]):
+    def forward(self, a_1: f32[s0, s1], b_1: f32[s2, s1]):
         # No stacktrace found for following nodes
-        sym_size: Sym(t0.size(0)) = torch.ops.aten.sym_size(a_1, 0)
-        sym_size_1: Sym(t1.size(0)) = torch.ops.aten.sym_size(b_1, 0)
-        add: Sym(t0.size(0) + t1.size(0)) = sym_size + sym_size_1;  sym_size = sym_size_1 = None
-        sym_size_2: Sym(t0.size(1)) = torch.ops.aten.sym_size(a_1, 1)
-        sym_size_3: Sym(t0.size(1)) = torch.ops.aten.sym_size(b_1, 1);  b_1 = None
-        add_1: Sym(2*t0.size(1)) = sym_size_2 + sym_size_3;  sym_size_2 = sym_size_3 = None
-        new_empty: f32[t0.size(0) + t1.size(0),2*t0.size(1)] = torch.ops.aten.new_empty.default(a_1, [add, add_1], dtype = torch.float32, layout = torch.strided, device = device(type='cpu'), pin_memory = False);  a_1 = add = add_1 = None
+        sym_size: Sym(s0) = torch.ops.aten.sym_size(a_1, 0)
+        sym_size_1: Sym(s2) = torch.ops.aten.sym_size(b_1, 0)
+        add: Sym(s0 + s2) = sym_size + sym_size_1;  sym_size = sym_size_1 = None
+        sym_size_2: Sym(s1) = torch.ops.aten.sym_size(a_1, 1)
+        sym_size_3: Sym(s1) = torch.ops.aten.sym_size(b_1, 1);  b_1 = None
+        add_1: Sym(2*s1) = sym_size_2 + sym_size_3;  sym_size_2 = sym_size_3 = None
+        new_empty: f32[s0 + s2, 2*s1] = torch.ops.aten.new_empty.default(a_1, [add, add_1], dtype = torch.float32, layout = torch.strided, device = device(type='cpu'), pin_memory = False);  a_1 = add = add_1 = None
         native_dropout = torch.ops.aten.native_dropout.default(new_empty, 0.5, True);  new_empty = None
-        getitem: f32[t0.size(0) + t1.size(0),2*t0.size(1)] = native_dropout[0]
-        getitem_1: b8[t0.size(0) + t1.size(0),2*t0.size(1)] = native_dropout[1];  native_dropout = None
+        getitem: f32[s0 + s2, 2*s1] = native_dropout[0]
+        getitem_1: b8[s0 + s2, 2*s1] = native_dropout[1];  native_dropout = None
         return (getitem, getitem_1)""")  # noqa: B950
 
 
diff --git a/test/test_dynamic_shapes.py.bak b/test/test_dynamic_shapes.py.bak
new file mode 100644
index 00000000000..19c77fe4d7a
--- /dev/null
+++ b/test/test_dynamic_shapes.py.bak
@@ -0,0 +1,391 @@
+# -*- coding: utf-8 -*-
+# Owner(s): ["oncall: jit"]
+
+from torch._C import _disabled_torch_function_impl
+import torch.fx
+import torch.nn.functional as F
+from torch.testing._internal.common_utils import run_tests, TestCase, skipIfTorchDynamo
+import unittest
+import torch
+import operator
+import itertools
+import io
+from torch.utils._pytree import tree_map
+from torch.fx.experimental.proxy_tensor import make_fx
+from torch.fx.experimental.symbolic_shapes import ShapeEnv, PySymInt, sym_float
+from torch.utils._python_dispatch import TorchDispatchMode
+
+aten = torch.ops.aten
+
+try:
+    import sympy
+    HAS_SYMPY = True
+except ImportError:
+    HAS_SYMPY = False
+skipIfNoSympy = unittest.skipIf(not HAS_SYMPY, "no sympy")
+
+
+meta_funcs = {}
+
+
+def register_meta(op):
+    def decorator(f):
+        def add_func(op):
+            meta_funcs[op] = f
+        tree_map(add_func, op)
+        return f
+    return decorator
+
+
+@register_meta([aten.add.Tensor, aten.sub.Tensor])
+def binary_meta(a, b):
+    return a.new_empty(a.shape)
+
+
+@register_meta(aten.cat.default)
+def cat_meta(tensors, dim=0):
+    concat_length = 0
+    shape = tensors[0].shape
+    for tensor in tensors:
+        for idx, (common_length, length) in enumerate(zip(shape, tensor.shape)):
+            if idx == dim:
+                concat_length = concat_length + length
+            else:
+                assert length == common_length
+    new_shape = list(shape)
+    new_shape[dim] = concat_length
+    return tensors[0].new_empty(new_shape)
+
+
+@register_meta([aten.narrow_copy.default])
+def narrow_copy_symint_meta(a, dim, start, length, **kwargs):
+    shape = []
+    for i, x in enumerate(a.shape):
+        if i == dim:
+            shape.append(length)
+        else:
+            shape.append(x)
+    return a.new_empty(tuple(shape))
+
+
+@register_meta([aten.expand.default])
+def expand_symint_meta(a, size, implicit=False):
+    return a.new_empty(size)
+
+
+def create_contiguous(shape):
+    strides = [1]
+    for dim in reversed(shape[:-1]):
+        strides.append(dim * strides[-1])
+    return list(reversed(strides))
+
+
+class FakeSymbolicTensor(torch.Tensor):
+    @staticmethod
+    def __new__(cls, sym_shape, sym_strides, dtype, layout, requires_grad, device, storage_offset=0):
+        # TODO: this is wrong in general
+        sym_stride = create_contiguous(sym_shape)
+        r = torch.Tensor._make_wrapper_subclass(
+            cls, sym_shape,
+            sym_stride, storage_offset,
+            dtype=dtype, layout=layout, requires_grad=requires_grad,
+            device=device,
+        )
+        return r
+
+    __torch_function__ = _disabled_torch_function_impl
+
+    def new_empty(self, shape):
+        return FakeSymbolicTensor(shape, None, self.dtype, self.layout, self.requires_grad, self.device)
+
+    @classmethod
+    def __torch_dispatch__(cls, func_overload, types, args=(), kwargs=None):
+        if func_overload in meta_funcs:
+            return meta_funcs[func_overload](*args, **kwargs)
+
+        if func_overload == torch.ops.aten.new_empty.default:
+            self = args[0]
+            shape = args[1]
+            return FakeSymbolicTensor(shape, self.stride(), self.dtype, self.layout, self.requires_grad, self.device)
+
+        raise RuntimeError(f"operator {func_overload} not supported")
+
+
+def create_symbolic_tensor(name, arg, shape_env, storage_offset=0):
+    sym_shapes, sym_strides = shape_env.create_symbolic_sizes_strides(arg)
+    return FakeSymbolicTensor(sym_shapes, sym_strides, arg.dtype, arg.layout, arg.requires_grad, arg.device, storage_offset)
+
+
+CPP_SYMINT_CLASS = type(torch.SymIntNode.new_symint(1))
+
+def create_symint(shape_env, i):
+    return shape_env.create_symintnode(shape_env.create_symbol(i))
+
+@skipIfTorchDynamo("Creating ShapeEnv fails for confusing reasons (also we never expect dynamo to see code like this)")
+class TestPySymInt(TestCase):
+
+    @skipIfNoSympy
+    def test_arith_ops(self):
+        shape_env = ShapeEnv()
+        symints = []
+        for i in range(2, 5):
+            symints.append((i, create_symint(shape_env, i)))
+
+        ops = [operator.add, operator.sub, operator.floordiv, operator.mul, operator.mod]
+
+        for op in ops:
+            for args in itertools.permutations(symints, 2):
+                if not isinstance(args[0][1], int) and ((op != operator.mod or op != operator.floordiv) and args[1][0] != 0):
+                    self.assertTrue(op(args[0][1], args[1][1]) == op(args[0][0], args[1][0]))
+
+
+    @skipIfNoSympy
+    def test_reverse_arith_ops(self):
+        shape_env = ShapeEnv()
+
+        a = create_symint(shape_env, 2)
+        self.assertTrue(5 // a == 5 // 2)
+
+        a = create_symint(shape_env, 2)
+        self.assertTrue(5 * a == 5 * 2)
+
+
+    @skipIfNoSympy
+    def test_roundtrip(self):
+        shape_env = ShapeEnv()
+        x = create_symbolic_tensor("x", torch.randn(5, 4, 3), shape_env)
+
+        self.assertTrue(not isinstance(x.shape[0], PySymInt))
+        self.assertTrue(isinstance(x.shape[0], CPP_SYMINT_CLASS))
+
+        self.assertTrue(x.shape[0] == 5)
+        self.assertTrue(x.shape[1] == 4)
+        self.assertTrue(x.shape[2], 3)
+
+        self.assertTrue(x.size()[0], 5)
+        self.assertTrue(x.size()[1], 4)
+        self.assertTrue(isinstance(x.size()[1], CPP_SYMINT_CLASS))
+        self.assertTrue(x.size()[2] == 3)
+
+        self.assertTrue(x.size(0) == 5)
+        self.assertTrue(x.size(1) == 4)
+        self.assertTrue(x.size(2) == 3)
+        self.assertTrue(isinstance(x.size(2), CPP_SYMINT_CLASS))
+
+        offset = create_symint(shape_env, 2)
+        y = create_symbolic_tensor("x", torch.randn(5, 4, 3), shape_env, offset)
+        self.assertTrue(isinstance(y.storage_offset(), CPP_SYMINT_CLASS))
+        self.assertTrue(y.storage_offset() == 2)
+
+        offset = 2
+        z = create_symbolic_tensor("z", torch.randn(5, 4, 3), shape_env, offset)
+        self.assertTrue(isinstance(z.storage_offset(), int))
+        self.assertTrue(z.storage_offset() == 2)
+
+    @skipIfNoSympy
+    def test_binary(self):
+        shape_env = ShapeEnv()
+        x = create_symbolic_tensor("x", torch.randn(5, 4, 3), shape_env)
+        y = create_symbolic_tensor("y", torch.randn(5, 4, 3), shape_env)
+
+        z = x + y
+        self.assertTrue(z.shape[0] == 5)
+        self.assertTrue(z.shape[1] == 4)
+        self.assertTrue(z.shape[2] == 3)
+
+        # broadcasting
+        y = create_symbolic_tensor("y", torch.randn(1, 4, 1), shape_env)
+        z = x + y
+        self.assertTrue(z.shape[0] == 5)
+        self.assertTrue(z.shape[1] == 4)
+        self.assertTrue(z.shape[2] == 3)
+
+    @skipIfNoSympy
+    def test_symint_args(self):
+        shape_env = ShapeEnv()
+        x = create_symbolic_tensor("x", torch.randn(5, 4, 3), shape_env)
+        y = create_symbolic_tensor("y", torch.randn(5, 4, 1), shape_env)
+        LAST_DIM = 2
+        z = x.narrow_copy(LAST_DIM, 0, y.shape[LAST_DIM])
+        self.assertTrue(z.shape[2] == y.shape[2])
+
+        # arithmetic expr with two symints
+        z = x.narrow_copy(LAST_DIM, 0, x.shape[LAST_DIM] - y.shape[LAST_DIM])
+        self.assertTrue(z.shape[2] == 2)
+
+        # arithmetic expr with a symint and python int
+        z = x.narrow_copy(LAST_DIM, 0, x.shape[LAST_DIM] - 1)
+        self.assertTrue(z.shape[2] == 2)
+
+    @skipIfNoSympy
+    def test_symint_vargs(self):
+        shape_env = ShapeEnv()
+        x = create_symbolic_tensor("x", torch.randn(5, 4, 3), shape_env)
+        y = create_symbolic_tensor("y", torch.randn(1, 4, 1), shape_env)
+
+        # varargs
+        z = y.expand(x.shape[0], y.shape[1], x.shape[2])
+        self.assertTrue(z.shape[0] == 5)
+        self.assertTrue(z.shape[1] == 4)
+        self.assertTrue(z.shape[2] == 3)
+
+        # shape list
+        z = y.expand((x.shape[0], y.shape[1], x.shape[2]))
+        self.assertTrue(z.shape[0] == 5)
+        self.assertTrue(z.shape[1] == 4)
+        self.assertTrue(z.shape[2] == 3)
+
+        # mixed python symints and ints
+        z = y.expand(x.shape[0], y.shape[1], 3)
+        self.assertTrue(z.shape[0] == 5)
+        self.assertTrue(z.shape[1] == 4)
+        self.assertTrue(z.shape[2] == 3)
+
+        # mixed python symints and ints in a list
+        z = y.expand((x.shape[0], y.shape[1], 3))
+        self.assertTrue(z.shape[0] == 5)
+        self.assertTrue(z.shape[1] == 4)
+        self.assertTrue(z.shape[2] == 3)
+
+        # mixed python symints and ints
+        z = y.expand(5, y.shape[1], x.shape[2])
+        self.assertTrue(z.shape[0] == 5)
+        self.assertTrue(z.shape[1] == 4)
+        self.assertTrue(z.shape[2] == 3)
+
+        # mixed python ints and symints in a list
+        z = y.expand((5, y.shape[1], x.shape[2]))
+        self.assertTrue(z.shape[0] == 5)
+        self.assertTrue(z.shape[1] == 4)
+        self.assertTrue(z.shape[2] == 3)
+
+        z = y.expand((y.shape[1],))
+        z = y.expand(y.shape[1])
+
+    @skipIfNoSympy
+    def test_stride(self):
+        shape_env = ShapeEnv()
+        x = create_symbolic_tensor("x", torch.randn(5, 5), shape_env)
+        self.assertIsInstance(x.stride()[0], CPP_SYMINT_CLASS)
+
+    @skipIfNoSympy
+    def test_size_expressions(self):
+        shape_env = ShapeEnv()
+        x = create_symbolic_tensor("x", torch.randn(5), shape_env)
+        expand_x = x.expand(x.shape[0], x.shape[0])
+        if expand_x.shape[0] > 3:
+            result = expand_x + expand_x
+        else:
+            result = expand_x + expand_x
+
+        gt_op = shape_env.guards[0][0]
+        self.assertTrue(isinstance(gt_op, sympy.core.relational.StrictGreaterThan))
+        self.assertTrue(str(x.shape[0]), str(gt_op.args[0]))
+        self.assertTrue(str(expand_x.shape[1]), str(x.shape[0]))
+        self.assertTrue(str(expand_x.shape[1]), str(result.shape[0]))
+
+    @skipIfNoSympy
+    def test_int_to_float(self):
+        shape_env = ShapeEnv()
+        x = create_symbolic_tensor("x", torch.randn(5), shape_env)
+        r = sym_float(x.shape[0])
+        self.assertTrue(isinstance(r, torch.SymFloatNode))
+
+    @skipIfNoSympy
+    def test_aten_ops(self):
+
+        shape_env = ShapeEnv()
+        x = create_symbolic_tensor("x", torch.randn(5), shape_env)
+        torch.ops.aten.narrow_copy.default(x, 0, 0, x.shape[0])
+
+        shape_env = ShapeEnv()
+        x = create_symbolic_tensor("x", torch.randn(5, 4, 3), shape_env)
+        torch.ops.aten.expand.default(x, [x.shape[0], x.shape[1], x.shape[2]])
+
+    def test_fx_trace_intlist(self):
+        class CustomModule(torch.nn.Module):
+            def forward(self, x):
+                bs, c, h, w = x.shape
+                return F.pad(x, (0, w % 2, 0, h % 2, 0, 0))
+
+        m = CustomModule()
+        x = torch.rand(1, 3, 4, 4)
+        # should not TypeError: pad(): argument 'pad' (position 2) must be
+        # tuple of ints, not tuple
+        torch.fx.symbolic_trace(m)
+
+    @skipIfNoSympy
+    def test_meta_symint(self):
+        shape_env = ShapeEnv()
+        a0 = create_symint(shape_env, 2)
+        r = torch.empty(a0, device='meta')
+        self.assertIsInstance(r.shape[0], CPP_SYMINT_CLASS)
+
+    @skipIfNoSympy
+    def test_guard_int(self):
+        shape_env = ShapeEnv()
+        a0 = create_symint(shape_env, 2)
+        self.assertEqual(a0.guard_int(), 2)
+        self.assertEqual(str(shape_env.guards[0][0]), "s0")
+        self.assertEqual(shape_env.guards[0][1], 2)
+
+    @skipIfNoSympy
+    def test_int_conversion(self):
+        shape_env = ShapeEnv()
+        a0 = create_symint(shape_env, 2)
+        self.assertRaisesRegex(RuntimeError, "Trying to extract", lambda: int(a0))
+
+    @skipIfNoSympy
+    def test_symint_as_scalar(self):
+        shape_env = ShapeEnv()
+        a0 = create_symint(shape_env, 2)
+
+        sym_int_encountered = False
+
+        class TestSymInt(TorchDispatchMode):
+            def __torch_dispatch__(self, func, types, args=(), kwargs=None):
+                assert func == torch.ops.aten.add.Tensor
+
+                nonlocal sym_int_encountered
+                sym_int_encountered = kwargs["alpha"] is a0
+                kwargs["alpha"] = 0
+                return func(*args)
+
+        x = torch.rand([4, 4])
+        with TestSymInt():
+            y = torch.add(x, x, alpha=a0)
+
+        self.assertTrue(sym_int_encountered)
+
+    @skipIfNoSympy
+    @unittest.mock.patch('sys.stdout', new_callable=io.StringIO)
+    def test_print_readable_with_symints(self, mock_stdout):
+        def f(a, b):
+            dim0 = a.shape[0] + b.shape[0]
+            dim1 = a.shape[1] + b.shape[1]
+            d = a.new_empty(dim0, dim1)
+            d = torch.ops.aten.native_dropout(d, 0.5, train=True)
+            return d
+
+        fx_g = make_fx(f, tracing_mode="symbolic")(torch.randn(5, 3), torch.randn(4, 3))
+        fx_g.print_readable()
+
+        self.assertExpectedInline(mock_stdout.getvalue().strip(), """\
+class f(torch.nn.Module):
+    def forward(self, a_1: f32[t0.size(0),t0.size(1)], b_1: f32[t1.size(0),t0.size(1)]):
+        # No stacktrace found for following nodes
+        sym_size: Sym(t0.size(0)) = torch.ops.aten.sym_size(a_1, 0)
+        sym_size_1: Sym(t1.size(0)) = torch.ops.aten.sym_size(b_1, 0)
+        add: Sym(t0.size(0) + t1.size(0)) = sym_size + sym_size_1;  sym_size = sym_size_1 = None
+        sym_size_2: Sym(t0.size(1)) = torch.ops.aten.sym_size(a_1, 1)
+        sym_size_3: Sym(t0.size(1)) = torch.ops.aten.sym_size(b_1, 1);  b_1 = None
+        add_1: Sym(2*t0.size(1)) = sym_size_2 + sym_size_3;  sym_size_2 = sym_size_3 = None
+        new_empty: f32[t0.size(0) + t1.size(0),2*t0.size(1)] = torch.ops.aten.new_empty.default(a_1, [add, add_1], dtype = torch.float32, layout = torch.strided, device = device(type='cpu'), pin_memory = False);  a_1 = add = add_1 = None
+        native_dropout = torch.ops.aten.native_dropout.default(new_empty, 0.5, True);  new_empty = None
+        getitem: f32[t0.size(0) + t1.size(0),2*t0.size(1)] = native_dropout[0]
+        getitem_1: b8[t0.size(0) + t1.size(0),2*t0.size(1)] = native_dropout[1];  native_dropout = None
+        return (getitem, getitem_1)""")  # noqa: B950
+
+
+if __name__ == '__main__':
+    run_tests()
diff --git a/test/test_proxy_tensor.py b/test/test_proxy_tensor.py
index d736a2c453a..4544171f6ef 100644
--- a/test/test_proxy_tensor.py
+++ b/test/test_proxy_tensor.py
@@ -1065,6 +1065,7 @@ symbolic_tensor_failures = {
     xfail('cholesky_solve', ''),  # Could not run 'aten::_cholesky_solve_helper' with arguments from the 'Meta' back...
     xfail('chunk', ''),  # aten.size.default - couldn't find symbolic meta function/decomposition
     xfail('column_stack', ''),  # Tensors of type TensorImpl do not have numel
+    xfail('combinations', ''),
     xfail('count_nonzero', ''),  # Could not run 'aten::count_nonzero.dim_IntList' with arguments from the 'Meta' ba...
     xfail('cross', ''),  # aten.linalg_cross.default - couldn't find symbolic meta function/decomposition
     xfail('cummax', ''),  # aten.cummax.default - couldn't find symbolic meta function/decomposition
@@ -1290,6 +1291,7 @@ symbolic_tensor_failures = {
     xfail('unbind', ''),  # aten.unbind.int - couldn't find symbolic meta function/decomposition
 }
 symbolic_tensor_segfaults = {
+    skip('nn.functional.batch_norm')  # Segfault??
 }
 
 symbolic_tensor_failures.update(symbolic_tensor_segfaults)
diff --git a/torch/_subclasses/fake_tensor.py b/torch/_subclasses/fake_tensor.py
index b9e1c867612..e5f70da1f19 100644
--- a/torch/_subclasses/fake_tensor.py
+++ b/torch/_subclasses/fake_tensor.py
@@ -622,6 +622,7 @@ class FakeTensorMode(TorchDispatchMode):
         allow_fallback_kernels=True,
         allow_meta=False,
         throw_on_data_dependent_ops=True,
+        shape_env=None,
     ):
         self.allow_fallback_kernels = allow_fallback_kernels
         self.fake_tensor_converter = FakeTensorConverter()
@@ -642,6 +643,8 @@ class FakeTensorMode(TorchDispatchMode):
         # the device property
         self.in_kernel_invocation = False
 
+        self.shape_env = shape_env
+
     def __torch_dispatch__(self, func, types, args=(), kwargs=None):
         kwargs = kwargs if kwargs else {}
 
@@ -920,8 +923,10 @@ class FakeTensorMode(TorchDispatchMode):
                 ):
                     self.fake_tensor_converter.invalidate_constant_aliases(v.constant)
 
-    def from_tensor(self, tensor, shape_env=None):
-        return self.fake_tensor_converter(self, tensor, shape_env=shape_env)
+    def from_tensor(self, tensor, static_shapes=False):
+        if static_shapes:
+            return self.fake_tensor_converter(self, tensor)
+        return self.fake_tensor_converter(self, tensor, shape_env=self.shape_env)
 
 
 # NB: returns fake tensors
diff --git a/torch/_subclasses/meta_utils.py b/torch/_subclasses/meta_utils.py
index 9d641ba458e..80723f12463 100644
--- a/torch/_subclasses/meta_utils.py
+++ b/torch/_subclasses/meta_utils.py
@@ -142,39 +142,18 @@ class MetaConverter:
         arg_cnt = self.arg_cnt
         self.arg_cnt += 1
 
-        # Don't make parameters have symbolic shapes; they are assumed to stay
-        # constant size across training runs
-        make_symbolic = shape_env is not None and not isinstance(t, torch.nn.Parameter)
+        make_symbolic = shape_env is not None
 
-        def sym(name, x):
+        def sym(x):
             if make_symbolic:
-                return shape_env.create_symint(f"t{arg_cnt}.{name}()", x)
+                return shape_env.create_symbol(x)
             else:
                 return x
 
-        def sym_list(name, xs):
+        def sym_sizes_strides(t):
             if make_symbolic:
-                return [
-                    shape_env.create_symint(f"t{arg_cnt}.{name}({i})", x)
-                    for i, x in enumerate(xs)
-                ]
-            else:
-                return xs
-
-        def sym_size(t):
-            return sym_list("size", t.size())
-
-        def sym_stride(t):
-            return sym_list("stride", t.stride())
-
-        # NB: Although sym_stride variables initially have no correlation
-        # with size, we will immediately introduce guards based on contiguity.
-        # Thus, if the input tensor is contiguous, the stride variables
-        # will typically immediately get reexpressed in terms of the size
-        # variables.
-
-        def sym_storage_offset(t):
-            return sym("storage_offset", t.storage_offset())
+                return shape_env.create_symbolic_sizes_strides(t)
+            return (t.size(), t.stride())
 
         # see expired-storages
         self.check_expired_count += 1
@@ -231,9 +210,8 @@ class MetaConverter:
                         base = base.view(t.dtype)
 
                     with torch.enable_grad():
-                        r = base.as_strided(
-                            sym_size(t), sym_stride(t), sym_storage_offset(t)
-                        )
+                        sizes, strides = sym_sizes_strides(t)
+                        r = base.as_strided(sizes, strides, sym(t.storage_offset()))
                 else:
                     is_leaf = safe_is_leaf(t)
                     # Fake up some autograd history.
@@ -257,8 +235,9 @@ class MetaConverter:
                     # meta storage
                     s = self.meta_storage(t.storage())
                     with no_dispatch():
+                        sizes, strides = sym_sizes_strides(t)
                         with torch.no_grad():
-                            r.set_(s, sym_storage_offset(t), sym_size(t), sym_stride(t))
+                            r.set_(s, sym(t.storage_offset()), sizes, strides)
 
                 torch._C._set_conj(r, t.is_conj())
                 torch._C._set_neg(r, t.is_neg())
diff --git a/torch/fx/experimental/proxy_tensor.py b/torch/fx/experimental/proxy_tensor.py
index de2cdaeb5b6..f623c4cfd65 100644
--- a/torch/fx/experimental/proxy_tensor.py
+++ b/torch/fx/experimental/proxy_tensor.py
@@ -612,7 +612,8 @@ def make_fx(f, decomposition_table=None, tracing_mode="real"):
         elif tracing_mode == "fake":
             fake_tensor_mode = FakeTensorMode(allow_fallback_kernels=True)
         elif tracing_mode == "symbolic":
-            fake_tensor_mode = FakeTensorMode(allow_fallback_kernels=False)
+            shape_env = ShapeEnv()
+            fake_tensor_mode = FakeTensorMode(allow_fallback_kernels=False, shape_env=shape_env)
         else:
             raise AssertionError(f"Unexpected tracing type: {tracing_mode}")
 
@@ -628,15 +629,12 @@ def make_fx(f, decomposition_table=None, tracing_mode="real"):
 
             return x
 
-        shape_env = None
-        if tracing_mode == "symbolic":
-            shape_env = ShapeEnv()
         sym_mode = proxy_mode.sym_mode
 
         # todo: Figure out a more informative name for symints
         def wrap_fake_symbolic(x):
             if isinstance(x, torch.Tensor):
-                return fake_tensor_mode.from_tensor(x, shape_env=shape_env)
+                return fake_tensor_mode.from_tensor(x)
             return x
 
         wrap_fn_map = {
diff --git a/torch/fx/experimental/symbolic_shapes.py b/torch/fx/experimental/symbolic_shapes.py
index 110831ddb9c..b42cfcfb109 100644
--- a/torch/fx/experimental/symbolic_shapes.py
+++ b/torch/fx/experimental/symbolic_shapes.py
@@ -1,6 +1,6 @@
 import torch
 import torch.utils._pytree as pytree
-from typing import Set, Dict, List, Type, Optional, cast
+from typing import Set, Dict, List, Type, Optional, cast, Union
 import operator
 import math
 import functools
@@ -331,7 +331,7 @@ class ShapeEnv(object):
         self.divisible: Set["sympy.Expr"] = set()
         # Duck-shaping says that if two input tensors have the same size,
         # they get assigned the same symbolic variable
-        self.val_to_symint: Dict[int, torch.SymIntNode] = {}
+        self.val_to_var: Dict[int, "sympy.Expr"] = {0: sympy.Integer(0), 1: sympy.Integer(1)}
 
     def _get_key(self):
         """
@@ -340,28 +340,68 @@ class ShapeEnv(object):
         """
         return (len(self.replacements), len(self.divisible))
 
-    # NB: This is only called for input symbolic sizes; intermediate symbolic
-    # sizes are allocated via a different mechanism
-    def create_symint(self, name, val):
-        assert val >= 0
+    def create_symbolic_sizes_strides(self, ex: torch.Tensor):
+        """
+        Returns a list of symbolic sizes and strides for the given tensor.
+        We try our best to express stride in terms of the sizes, so as to not
+        introduce new symbolic variables.
+        """
+
+        size = [self.create_symbol(i) for i in ex.size()]
+        stride: List[Optional[sympy.Expr]] = [None] * len(size)
+        for i, val in enumerate(ex.stride()):
+            if val in (0, 1):
+                stride[i] = sympy.Integer(val)
+        while any(x is None for x in stride):
+            candidates = {
+                ex.size(i) * ex.stride()[i]: size[i] * stride[i]
+                for i in range(len(size))
+                if stride[i] is not None and ex.stride()[i] >= 0
+            }
+            # iterate over unbound strides in sorted order
+            val_list = sorted(
+                [(ex.stride()[i], i) for i in range(len(stride)) if stride[i] is None]
+            )
+            for _, i in val_list:
+                if stride[i] is None and ex.stride()[i] in candidates:
+                    stride[i] = candidates[ex.stride()[i]]
+                    candidates[ex.size(i) * ex.stride()[i]] = size[i] * stride[i]
+            if any(x is None for x in stride):
+                # bind the smallest unbound stride to a new variable
+                val, i = sorted(
+                    [
+                        (ex.stride()[i], i)
+                        for i in range(len(stride))
+                        if stride[i] is None
+                    ]
+                )[0]
+                stride[i] = self.create_symbol(val)
+        assert all(x is not None for x in stride)
+        return [self.create_symintnode(i) for i in size], [self.create_symintnode(i) for i in stride]  # type: ignore[arg-type]
+
+    def create_symintnode(self, expr: Union["sympy.Expr", int]):
+        py_sym_int = PySymInt(expr, self)
+        cpp_sym_int = torch.SymIntNode.new_symint(py_sym_int)  # type: ignore[attr-defined]
+        return cpp_sym_int
+
+    def create_symbol(self, val: int) -> "sympy.Expr":
         if not HAS_SYMPY:
             raise RuntimeError("Need sympy installed to create symbolic shapes")
-
-        # TODO: Put 0/1 specialization in guards
-        if val == 0 or val == 1:
-            return val
+        if val < 0:
+            # all sympy base variables must be positive and > 1
+            return -self.create_symbol(-val)
         # This implements duck-shaping: input sizes that match are assigned
         # the same symint
         # TODO: Create a guard whenever this happens
         # TODO: But how do I represent the guard in this case?
-        if val in self.val_to_symint:
-            return self.val_to_symint[val]
-        sympy_expr = sympy.Symbol(name, positive=True, integer=True)
-        py_sym_int = PySymInt(sympy_expr, self)
-        cpp_sym_int = torch.SymIntNode.new_symint(py_sym_int)  # type: ignore[attr-defined]
+        # Note: val_to_var is also initialized with 0/1 mapping to constants, so
+        # this also ensures that all symbols are > 1
+        if val in self.val_to_var:
+            return self.val_to_var[val]
+        sympy_expr = sympy.Symbol(f"s{len(self.var_to_val)}", positive=True, integer=True)
         self.var_to_val[sympy_expr] = sympy.Integer(val)
-        self.val_to_symint[val] = cpp_sym_int
-        return cpp_sym_int
+        self.val_to_var[val] = sympy_expr
+        return sympy_expr
 
     def evaluate_guards_for_args(self, *args):
         new_env = ShapeEnv()
diff --git a/torch/fx/graph.py b/torch/fx/graph.py
index b5e21f1c3de..4aa6897196d 100644
--- a/torch/fx/graph.py
+++ b/torch/fx/graph.py
@@ -475,7 +475,7 @@ class CodeGen(object):
                     body.append('\n# No stacktrace found for following nodes\n')
 
         def stringify_shape(shape : torch.Size) -> str:
-            return f"[{','.join(str(x) for x in shape)}]"
+            return f"[{', '.join(str(x) for x in shape)}]"
 
         def emit_node(node : Node):
             maybe_type_annotation = '' if node.type is None else f' : {type_repr(node.type)}'